#72321817

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#13685
commutator, Frattini, cosocle	C₂×C₃₂	C₂⁴
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ #13549 ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 128 points:

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

Polycyclic presentation:

2^10|b^2=fj,e^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bf,c^a=ce,c^b=ci,d^b=dj,e^a=eg,e^c=eg,f^c=fj,g^a=gh,g^c=gh,h^a=hi,h^c=hi,i^a=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321842, #72322926, #72322965
maximal normal subgroup of	#72321842, #72322926, #72322965

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, 2, F.6*F.10);;
gap> SetPower(coll, 5, 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, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
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);