#72268601

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁸
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

70044635 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 8 generators on 24 points:

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

Polycyclic presentation:

2^12|g^2=l,h^2=ik,g^b=gl,g^c=gk,g^e=gj,g^f=gi,h^a=hk,h^b=hk,h^d=hi,h^e=hi

Conjugacy classes & character table

640 conjugacy classes — view the full character table.

Relationships

automorphism group of

#3536, #3558, #3559, …

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", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 7, F.12);;
gap> SetPower(coll, 8, F.9*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);