#72309550

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	#72309533	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #183995 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^11|a^2=f,b^2=k,c^2=i,d^2=f,e^2=h,g^2=i,h^2=j,j^2=k,c^a=ck,c^b=cg,d^a=dk,d^c=di,e^a=eh,e^d=ej,f^e=fk,g^b=gi,g^c=gi,h^a=hj,h^d=hk,j^a=jk

Conjugacy classes & character table

131 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, 1, F.6);;
gap> SetPower(coll, 2, F.11);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 5, 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, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);