#72231968

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#346
prime radical	C₂×#41	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

631 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:

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

Polycyclic presentation:

2^10|a^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=j,c^a=cj,c^b=chij,d^a=dhj,d^b=dj,e^d=ej,f^a=fj

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231969
maximal normal subgroup of	#72231969

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