#72231911

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

1070 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)(17,18,20,28,22,30,44,62,24,32,45,64,23,54,21,48,26,34,47,66,53,68,49,29,25,33,46,65,52,31,50,63)(19,35,37,70,39,55,73,80,41,67,76,79,74,60,77,61,36,69,72,57,51,27,38,56,42,58,43,78,40,59,75,71);(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)(18,27,29,57,31,35,66,71,33,60,48,79,68,58,64,56,34,55,62,70,54,69,28,61,32,59,63,78,30,67,65,80);(2,7)(6,12)(8,14)(13,15);(19,36)(27,55)(35,69)(37,72)(38,73)(39,51)(40,74)(41,42)(43,76)(56,80)(57,70)(58,67)(59,60)(61,71)(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^b=cij,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	#72231918, #72322903
maximal normal subgroup of	#72231918, #72322903

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, 2, F.3*F.9*F.10);;
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);