2₋¹⁺¹⁰ #72258204

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂¹⁰ · Schur multiplier (H₂) = C₂⁴⁴

Aliases

extraspecial

2₋¹⁺¹⁰

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂	C₂¹⁰
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ G
LECS	G ▷ C₂ ▷ C₁

229876716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 10 generators on 128 points:

(1,2,8,14)(3,15,20,9)(4,10,21,35)(5,11,25,39)(6,12,28,42)(7,13,30,44)(16,45,49,31)(17,46,53,32)(18,47,56,33)(19,48,58,34)(22,36,59,75)(23,37,60,76)(24,38,61,77)(26,40,65,81)(27,41,66,82)(29,43,68,84)(50,85,91,69)(51,86,92,70)(52,87,93,71)(54,88,97,72)(55,89,98,73)(57,90,100,74)(62,78,101,110)(63,79,102,111)(64,80,103,112)(67,83,105,114)(94,115,119,106)(95,116,120,107)(96,117,121,108)(99,118,123,109)(104,113,124,126)(122,127,128,125);(1,3,8,20)(2,9,14,15)(4,16,21,49)(5,17,25,53)(6,18,28,56)(7,19,30,58)(10,31,35,45)(11,32,39,46)(12,33,42,47)(13,34,44,48)(22,50,59,91)(23,51,60,92)(24,52,61,93)(26,54,65,97)(27,55,66,98)(29,57,68,100)(36,69,75,85)(37,70,76,86)(38,71,77,87)(40,72,81,88)(41,73,82,89)(43,74,84,90)(62,94,101,119)(63,95,102,120)(64,96,103,121)(67,99,105,123)(78,106,110,115)(79,107,111,116)(80,108,112,117)(83,109,114,118)(104,122,124,128)(113,125,126,127);(4,21)(10,35)(16,49)(22,59)(23,60)(24,61)(31,45)(36,75)(37,76)(38,77)(50,91)(51,92)(52,93)(62,101)(63,102)(64,103)(69,85)(70,86)(71,87)(78,110)(79,111)(80,112)(94,119)(95,120)(96,121)(104,124)(106,115)(107,116)(108,117)(113,126)(122,128)(125,127);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,21)(9,31)(11,36)(12,37)(13,38)(14,35)(15,45)(17,50)(18,51)(19,52)(20,49)(25,59)(26,62)(27,63)(28,60)(29,64)(30,61)(32,69)(33,70)(34,71)(39,75)(40,78)(41,79)(42,76)(43,80)(44,77)(46,85)(47,86)(48,87)(53,91)(54,94)(55,95)(56,92)(57,96)(58,93)(65,101)(66,102)(67,104)(68,103)(72,106)(73,107)(74,108)(81,110)(82,111)(83,113)(84,112)(88,115)(89,116)(90,117)(97,119)(98,120)(99,122)(100,121)(105,124)(109,125)(114,126)(118,127)(123,128);(5,25)(11,39)(17,53)(22,59)(26,65)(27,66)(32,46)(36,75)(40,81)(41,82)(50,91)(54,97)(55,98)(62,101)(63,102)(67,105)(69,85)(72,88)(73,89)(78,110)(79,111)(83,114)(94,119)(95,120)(99,123)(104,124)(106,115)(107,116)(109,118)(113,126)(122,128)(125,127);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,25)(9,32)(10,36)(12,40)(13,41)(14,39)(15,46)(16,50)(18,54)(19,55)(20,53)(21,59)(23,62)(24,63)(28,65)(29,67)(30,66)(31,69)(33,72)(34,73)(35,75)(37,78)(38,79)(42,81)(43,83)(44,82)(45,85)(47,88)(48,89)(49,91)(51,94)(52,95)(56,97)(57,99)(58,98)(60,101)(61,102)(64,104)(68,105)(70,106)(71,107)(74,109)(76,110)(77,111)(80,113)(84,114)(86,115)(87,116)(90,118)(92,119)(93,120)(96,122)(100,123)(103,124)(108,125)(112,126)(117,127)(121,128);(6,28)(12,42)(18,56)(23,60)(26,65)(29,68)(33,47)(37,76)(40,81)(43,84)(51,92)(54,97)(57,100)(62,101)(64,103)(67,105)(70,86)(72,88)(74,90)(78,110)(80,112)(83,114)(94,119)(96,121)(99,123)(104,124)(106,115)(108,117)(109,118)(113,126)(122,128)(125,127);(1,6)(2,12)(3,18)(4,23)(5,26)(7,29)(8,28)(9,33)(10,37)(11,40)(13,43)(14,42)(15,47)(16,51)(17,54)(19,57)(20,56)(21,60)(22,62)(24,64)(25,65)(27,67)(30,68)(31,70)(32,72)(34,74)(35,76)(36,78)(38,80)(39,81)(41,83)(44,84)(45,86)(46,88)(48,90)(49,92)(50,94)(52,96)(53,97)(55,99)(58,100)(59,101)(61,103)(63,104)(66,105)(69,106)(71,108)(73,109)(75,110)(77,112)(79,113)(82,114)(85,115)(87,117)(89,118)(91,119)(93,121)(95,122)(98,123)(102,124)(107,125)(111,126)(116,127)(120,128);(7,30)(13,44)(19,58)(24,61)(27,66)(29,68)(34,48)(38,77)(41,82)(43,84)(52,93)(55,98)(57,100)(63,102)(64,103)(67,105)(71,87)(73,89)(74,90)(79,111)(80,112)(83,114)(95,120)(96,121)(99,123)(104,124)(107,116)(108,117)(109,118)(113,126)(122,128)(125,127);(1,7)(2,13)(3,19)(4,24)(5,27)(6,29)(8,30)(9,34)(10,38)(11,41)(12,43)(14,44)(15,48)(16,52)(17,55)(18,57)(20,58)(21,61)(22,63)(23,64)(25,66)(26,67)(28,68)(31,71)(32,73)(33,74)(35,77)(36,79)(37,80)(39,82)(40,83)(42,84)(45,87)(46,89)(47,90)(49,93)(50,95)(51,96)(53,98)(54,99)(56,100)(59,102)(60,103)(62,104)(65,105)(69,107)(70,108)(72,109)(75,111)(76,112)(78,113)(81,114)(85,116)(86,117)(88,118)(91,120)(92,121)(94,122)(97,123)(101,124)(106,125)(110,126)(115,127)(119,128)

Polycyclic presentation:

2^11|a^2=k,b^2=k,b^a=bk,d^c=dk,f^e=fk,h^g=hk,j^i=jk

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×2₋¹⁺¹⁰, #72274291
maximal normal subgroup of	C₂×2₋¹⁺¹⁰, #72274291
maximal quotient of	C₂×2₋¹⁺¹⁰

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.11);;
gap> SetPower(coll, 2, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 10, 9, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

2−1+10 #72258204