#72225282

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#21690
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
socle	C₂²	C₂×#21690
prime radical	#72225180	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#7846 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5498 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:

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

Polycyclic presentation:

2^11|a^2=f,b^2=fh,c^2=g,d^2=k,e^2=hj,f^2=j,g^2=i,h^2=j,i^2=k,b^a=bh,c^a=cg,c^b=ci,e^a=ehj,e^b=ej,f^c=fk,g^a=gi,g^b=gk,h^a=hj,i^a=ik

Conjugacy classes & character table

176 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.6);;
gap> SetPower(coll, 2, F.6*F.8);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.11);;
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, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
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);