#72300754

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄

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

At least 8218 conjugacy classes of subgroups; only the 3129 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|b^2=ij,c^2=ij,d^2=h,e^2=ij,b^a=bi,d^b=dg,d^c=di,e^d=ehij,f^a=fh,f^b=fgh,f^d=fh,f^e=fgh

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300808
maximal normal subgroup of	#72300808

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