#72319994

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	D₄×#442
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#72319413	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #184045 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4523 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)(21,36)(22,27)(23,38)(24,28)(29,39)(30,45)(31,41)(32,46)(33,42)(34,47)(35,43)(37,48)(40,44);(2,6,8,15)(3,9)(4,12)(7,10,13,16)(11,14)(17,19,22,31,24,33,23,32)(18,25,27,41,28,42,38,46)(20,29,21,37,35,44,34,30)(26,39,36,48,43,40,47,45);(1,2)(3,10)(4,13)(5,8)(6,14)(7,12)(9,15)(11,16);(2,7,8,13)(3,11)(6,10,15,16)(9,14)(17,20,23,34,24,35,22,21)(18,26,38,47,28,43,27,36)(19,29,32,30,33,44,31,37)(25,39,46,45,42,40,41,48);(17,21,22,35,24,34,23,20)(18,26,38,47,28,43,27,36)(19,30,31,29,33,37,32,44)(25,40,46,48,42,39,41,45)

Polycyclic presentation:

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

Conjugacy classes & character table

137 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, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.8*F.10);;
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, 4, 3, F.4*F.9*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, 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*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
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);