#72250056

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346518
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,9,15)(3,16,7,10)(4,14,8,11)(5,23,26,42)(6,27,28,13)(12,24,41,25)(17,33,20,36)(18,47,29,57)(19,35,30,43)(21,53,31,58)(22,38,32,44)(34,49,39,51)(37,54,40,55)(45,64,50,62)(46,56,52,61)(48,63,59,60);(1,3,9,7)(2,10,15,16)(4,17,8,20)(5,24)(6,19,28,30)(11,33,14,36)(12,23)(13,35,27,43)(18,48)(21,54)(22,46,32,52)(25,26)(29,59)(31,55)(34,62)(37,53)(38,61,44,56)(39,64)(40,58)(41,42)(45,51)(47,60)(49,50)(57,63);(1,4,9,8)(2,11,15,14)(3,17,7,20)(5,24)(6,22,28,32)(10,33,16,36)(12,23)(13,38,27,44)(18,49)(19,46,30,52)(21,55)(25,26)(29,51)(31,54)(34,47)(35,61,43,56)(37,58)(39,57)(40,53)(41,42)(45,59)(48,50)(60,62)(63,64);(1,5,17,45,4,21,3,18,9,26,20,50,8,31,7,29)(2,12,33,60,14,40,16,39,15,41,36,63,11,37,10,34)(6,24,46,48,22,55,19,51,28,25,52,59,32,54,30,49)(13,42,61,62,44,58,43,57,27,23,56,64,38,53,35,47)

Polycyclic presentation:

2^10|a^2=j,b^2=hj,c^2=hj,d^2=fgj,e^2=j,f^2=ij,g^2=j,i^2=j,b^a=bj,c^a=cj,d^a=de,d^b=defi,d^c=degj,e^a=ej,e^b=eh,e^c=eh,e^d=ej,f^a=fh,f^b=fhij,f^c=fhj,f^d=fj,f^e=fj,g^a=gh,g^b=ghj,g^c=gh,g^d=gj,g^e=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250237
maximal normal subgroup of	#72250237
maximal quotient of	#72250571, #72280512

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