#6481978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389174);