#72336513

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

28060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|f^2=gij,b^a=bh,c^a=cj,c^b=ci,d^c=dg,e^c=eh,f^a=fj,f^c=fgh,f^d=fhij,f^e=fgh

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336654
maximal normal subgroup of	#72336654

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