#72263573

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

24469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=i,b^2=h,c^2=ghi,d^2=j,b^a=bh,c^a=cg,d^c=dhi,e^a=eij,e^c=ehj,e^d=ej,f^a=fj,f^c=fhj

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263574, #72263630
maximal normal subgroup of	#72263574, #72263630

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