#72265685

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#368117
commutator	C₂×C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
prime radical	#7623353	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #355237 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4372 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:

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

Polycyclic presentation:

2^11|a^2=f,b^2=f,c^2=h,d^2=efgijk,e^2=ik,g^2=j,j^2=k,c^a=ch,c^b=ch,d^a=dhj,d^b=dg,d^c=de,e^a=ei,e^b=ek,e^c=ek,e^d=ek,f^d=fk,g^a=gk,g^b=gj,g^c=gik,h^d=hi,j^b=jk

Conjugacy classes & character table

104 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, 2, F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9*F.10*F.11);;
gap> SetPower(coll, 5, F.9*F.11);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);