#72287137

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#33758	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=h,c^2=f,d^2=g,f^2=h,g^2=ij,i^2=j,c^a=chj,c^b=cg,d^a=dhj,d^b=dgij,d^c=dg,e^b=eh,e^c=ej,e^d=ej,f^b=fj,f^d=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287214, #72287790, #72287829
maximal normal subgroup of	#72287214, #72287790, #72287829

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