#72266534

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

At least 7811 conjugacy classes of subgroups; only the 3013 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)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,26,28,30)(27,31,32,29);(2,5)(6,8)(9,11)(10,12)(13,14)(15,16);(9,12,13,16)(10,14,15,11)(17,19)(18,22)(20,23)(21,24)(27,32)(29,31);(11,14)(12,16)(19,24)(20,22)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,27)(26,29)(28,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^10|a^2=i,c^2=g,c^a=cj,d^a=di,d^b=dg,d^c=dj,e^a=ei,e^b=eh,e^c=eij,f^a=fhi,f^c=fj,f^d=fj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266546, #72266583, #72266584
maximal normal subgroup of	#72266546, #72266583, #72266584
maximal quotient of	#72334886

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