#72321377

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂³×D₃₂
commutator, Frattini, cosocle	C₃₂	C₂⁵
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ C₂×#2258 ◁ G
LECS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 128 points:

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

Polycyclic presentation:

2^10|f^2=gh,g^2=hi,h^2=ij,i^2=j,c^a=cf,c^b=cj,d^b=dj,e^c=ej,f^a=fg,f^c=fg,g^a=gh,g^c=gh,h^a=hi,h^c=hi,i^a=ij,i^c=ij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321422, #72322943, #72322947
maximal normal subgroup of	#72321422, #72322943, #72322947

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);