#72336046

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

At least 7270 conjugacy classes of subgroups; only the 3829 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:

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

Polycyclic presentation:

2^10|a^2=i,d^2=gij,e^2=g,f^2=gj,b^a=bg,c^a=cij,d^a=dgj,d^c=dij,e^a=egi,e^c=eij,e^d=eij,f^a=fg,f^d=fgh,f^e=fhj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336073
maximal normal subgroup of	#72336073
maximal quotient of	#72336469

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