#72331243

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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 10243 conjugacy classes of subgroups; only the 3777 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=i,d^2=h,e^2=g,b^a=bj,c^b=ci,d^b=dgh,d^c=dhi,e^c=egi,e^d=egh,f^a=fi,f^c=fi,f^d=fgh

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331261
maximal normal subgroup of	#72331261
maximal quotient of	#72333339

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