#6466885

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=ghi,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6374081);