#72231971

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

1064 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,j^2=k,b^a=bh,c^a=ck,c^b=cijk,d^b=dk,e^b=ej,f^a=fj,f^e=fk,g^b=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, 2, 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, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, 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, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);