#72231910

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×C₄	C₂²×#322
commutator	C₂×C₈	C₂²×C₄×C₈
Frattini, cosocle	#1311	C₂⁴
socle	C₂²	C₂²×#1313
prime radical	C₂×#41	#322
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #479899 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231918, #72231972, #72323133
maximal normal subgroup of	#72231918, #72231972, #72323133

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, 3, 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, 3, 1, F.3*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, 3, 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, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);