#72321033

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	#72320531	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

4530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^11|b^2=f,d^2=fj,e^2=h,f^2=j,g^2=i,h^2=j,i^2=k,c^b=cg,d^c=dik,e^a=eh,e^c=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);;
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, 3, 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);