#72267679

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#120	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

16307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,6,5)(3,8,7,4)(9,10,12,14)(11,13,15,16)(18,22)(20,24)(25,26,28,30)(27,29,31,32);(1,3,6,7)(2,4,5,8)(11,15)(13,16)(18,22)(19,23)(25,27,28,31)(26,29,30,32);(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,29)(28,30)(31,32);(9,11)(10,13)(12,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32);(1,4,6,8)(2,7,5,3)(9,11)(10,13)(12,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,29)(28,31)(30,32);(1,5,6,2)(3,4,7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^10|a^2=g,b^2=gh,e^2=j,f^2=j,b^a=bj,c^a=ci,c^b=cgij,d^a=dghj,d^b=dghij,e^a=ej,e^b=eij,f^a=fi,f^b=fj,f^e=fj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267687

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