#72231964

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×#322
commutator	C₂×C₈	C₂²×C₄×C₈
Frattini, cosocle	#1311	C₂⁴
socle	C₂²	C₂²×#1313
prime radical	C₂⁴	#1313
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(17,18)(19,29,31,52,30,51,34,54)(20,35,36,64)(21,39)(22,28)(23,24)(25,33,46,59,43,56,48,63)(26,38,44,57)(27,41)(32,60,61,77,55,78,58,53)(37,68,67,66)(40,71)(42,45)(47,75,65,79,73,80,74,62)(49,76,70,69)(50,72);(2,6)(7,12)(8,13)(14,16)(19,30)(25,43)(29,51)(31,34)(32,55)(33,56)(46,48)(47,73)(52,54)(53,77)(58,61)(59,63)(60,78)(62,79)(65,74)(75,80);(18,24)(25,43)(26,44)(27,39)(28,45)(29,51)(35,64)(46,48)(47,73)(49,70)(50,71)(52,54)(53,77)(60,78)(65,74)(66,68);(17,19,20,32,21,33,37,62,22,34,38,61,40,63,69,75,23,30,36,55,41,56,67,79,42,31,57,58,72,59,76,80)(18,25,26,47,27,29,49,53,28,48,35,65,50,54,66,60,24,43,44,73,39,51,70,77,45,46,64,74,71,52,68,78)

Polycyclic presentation:

2^11|a^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bi,c^a=ck,d^a=dh,d^b=dk,e^d=ej,f^a=fj,f^e=fk,g^d=gk,h^a=hk

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231972
maximal normal subgroup of	#72231972

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