#6775665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

1423 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,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=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6682861);