#72262829

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

15086 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:

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

Polycyclic presentation:

2^10|b^2=h,d^2=h,e^2=j,b^a=bh,c^b=cg,d^c=di,e^a=egi,e^c=eij,e^d=ehj,f^a=fi,f^b=fh,f^d=fj,f^e=fj

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262832, #72262846
maximal normal subgroup of	#72262832, #72262846

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