#6566953

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1275 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,32,29)(27,34,37,39)(28,38,36,33)(30,35,31,40);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,36,32,37,30,28)(26,33,35,34,29,38,40,39);(1,3,5,4)(2,7,8,6)(11,17)(14,20)(18,23)(21,24)(25,28,31,27,32,36,30,37)(26,34,35,38,29,39,40,33);(1,4,5,3)(2,6,8,7)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,29,32,26)(27,33,37,38)(28,39,36,34)(30,35,31,40);(12,19)(15,22)(18,23)(21,24)(25,30,32,31)(26,35,29,40)(27,28,37,36)(33,34,38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh

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, 6474149);