#72321818

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×#106
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	C₂×#575	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₈×#575 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(17,18)(19,25)(20,26)(21,34)(22,28)(23,38)(24,29)(27,37)(30,44)(31,40)(32,41)(33,39)(35,47)(36,43)(42,45)(46,48);(2,6,8,14)(3,10)(4,7)(9,16,13,11)(12,15)(17,19,22,31,24,32,23,20)(18,25,28,40,29,41,38,26)(21,30,36,46,35,45,37,33)(27,39,34,44,43,48,47,42);(1,2,7,9,5,8,4,13)(3,11,15,14,12,16,10,6)(17,20,23,32,24,31,22,19)(18,26,38,41,29,40,28,25)(21,33,37,45,35,46,36,30)(27,42,47,48,43,44,34,39);(21,35)(27,43)(30,45)(33,46)(34,47)(36,37)(39,48)(42,44);(17,21,23,37,24,35,22,36)(18,27,28,34,29,43,38,47)(19,30,20,33,32,45,31,46)(25,39,40,44,41,48,26,42)

Polycyclic presentation:

2^11|b^2=f,c^2=fijk,e^2=h,f^2=j,g^2=i,h^2=j,i^2=k,c^b=cg,e^a=eh,e^d=ej,f^c=fk,g^b=gi,g^c=gi,h^a=hj,i^b=ik,i^c=ik

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

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