#72268600

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁷
commutator	C₂³	C₂⁶×C₄
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2753392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 24 points:

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

Polycyclic presentation:

2^11|e^2=i,g^2=hj,e^b=ek,e^c=ej,f^e=fh,g^a=gj,g^b=gj,g^d=gh,g^e=gh

Conjugacy classes & character table

416 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268601
maximal normal subgroup of	#72268601

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