#72305960

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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

15534 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18)(19,24)(20,22)(21,23)(25,26)(27,30)(28,31)(29,32);(1,2,5,6)(3,7,8,4)(9,11,13,14)(10,12,15,16)(17,18)(19,24)(20,22)(21,23)(25,27)(26,30)(28,32)(29,31);(2,6)(3,8)(9,10)(11,12)(13,15)(14,16)(25,27)(26,30)(28,32)(29,31);(1,3,5,8)(2,7,6,4)(9,12,13,16)(10,11,15,14)(17,19,21,22)(18,20,23,24)(25,26,29,32)(27,28,31,30);(1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32);(1,3)(2,7)(4,6)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,28)(26,31)(27,32)(29,30)

Polycyclic presentation:

2^10|b^2=i,d^2=j,b^a=bi,c^a=cg,c^b=cg,d^a=dgj,d^b=dij,d^c=dh,e^a=eh,f^c=fi,f^d=fi,f^e=fj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305966, #72320656
maximal normal subgroup of	#72305966, #72320656

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