#6512605

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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

1314 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6419801);