#6775035

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(17,18,22,27)(19,29,31,24)(20,26,21,34)(23,38,37,28)(25,36,42,33)(30,40,35,43)(32,41,47,45)(39,46,44,48);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,35)(22,31)(23,32)(25,39)(26,43)(27,29)(28,41)(33,46)(34,40)(36,48)(37,47)(38,45)(42,44);(10,13)(14,16)(17,20,22,21)(18,25,27,42)(19,30,31,35)(23,36,37,33)(24,39,29,44)(26,28,34,38)(32,48,47,46)(40,45,43,41);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,22,20)(18,26,27,34)(19,30,31,35)(23,36,37,33)(24,40,29,43)(25,28,42,38)(32,46,47,48)(39,45,44,41);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,31)(20,21)(24,29)(25,42)(26,34)(32,47)(33,36)(41,45)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6682231);