#6354364

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354359
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1758 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,21,34)(12,17,26,33)(13,18,14,28)(16,23,29,24)(22,35,36,40)(25,31,27,32)(30,38,39,37);(9,11,13,23,15,21,14,24)(10,16,28,34,19,29,18,20)(12,22,27,38,26,36,25,37)(17,30,31,40,33,39,32,35);(2,5)(3,8)(6,7)(11,21)(16,29)(20,34)(22,36)(23,24)(30,39)(35,40)(37,38);(2,6)(5,7)(9,12)(10,17)(11,22)(13,27)(14,25)(15,26)(16,30)(18,32)(19,33)(20,35)(21,36)(23,38)(24,37)(28,31)(29,39)(34,40);(9,13,15,14)(10,18,19,28)(11,23,21,24)(12,25,26,27)(16,20,29,34)(17,31,33,32)(22,37,36,38)(30,40,39,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261560);