#72308246

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#364376
commutator	C₂×C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
prime radical	#72265737	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #355508 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5881 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^11|a^2=f,c^2=e,d^2=fghj,e^2=ij,g^2=jk,h^2=jk,j^2=k,c^a=cg,c^b=cghj,d^a=dghj,d^b=dghj,d^c=degik,e^a=ej,e^b=ei,e^d=eik,f^c=fk,g^a=gjk,g^c=gk,g^d=gi,h^a=hjk,h^c=hik,h^d=hi,j^a=jk

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308663
maximal normal subgroup of	#72308663

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