#6357322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357317
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2195 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ci,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, 6264518);