#72331471

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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 11764 conjugacy classes of subgroups; only the 4237 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=i,b^a=bj,c^a=ci,c^b=ch,d^a=dg,f^b=fh,f^c=fi,f^d=fhi,f^e=fh

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331604
maximal normal subgroup of	#72331604
maximal quotient of	#72333201

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