#6466089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373285);