#72226953

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#212918
commutator	C₈²	C₂³×C₄
Frattini, cosocle	C₂×C₈²	C₂⁴
prime radical	#72225197	C₂
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #7846 ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

1820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^11|a^2=e,b^2=ek,c^2=fk,d^2=g,f^2=h,g^2=i,h^2=j,i^2=k,c^a=cf,c^b=ch,d^a=dg,d^b=di,e^c=ej,e^d=ek,f^a=fh,f^b=fj,g^a=gi,g^b=gk,h^a=hj,i^a=ik

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226962
maximal normal subgroup of	#72226962

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