#72280926

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×#127
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁵
socle	C₂³	C₂²×#392
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁶×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

553539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^11|a^2=k,c^2=h,d^2=gi,f^2=k,c^a=ch,c^b=ch,d^a=dg,d^b=dgi,d^c=df,e^d=ej,g^c=gk,h^d=hk

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303605
maximal normal subgroup of	#72303605

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