#6466564

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1886 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)(3,7)(4,5)(6,8)(9,10)(11,22)(12,17)(13,18)(14,19)(15,21)(16,25)(20,31)(23,36)(24,32)(26,35)(27,34)(28,40)(29,38)(30,37)(33,39);(2,5)(6,7)(9,11,14,25,15,27,31,26)(10,16,20,22,21,35,19,34)(12,28,13,23,29,24,30,39)(17,36,37,40,38,33,18,32);(1,3,4,8)(2,6,5,7)(9,12,14,13,15,29,31,30)(10,17,19,18,21,38,20,37)(11,23,26,28,27,39,25,24)(16,32,22,36,35,40,34,33);(9,13,14,29,15,30,31,12)(10,18,19,38,21,37,20,17)(11,24,26,23,27,28,25,39)(16,33,22,32,35,36,34,40);(9,14,15,31)(10,19,21,20)(11,25,27,26)(12,13,29,30)(16,34,35,22)(17,18,38,37)(23,24,39,28)(32,33,40,36)

Polycyclic presentation:

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

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, 6373760);