#6362634

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ehi,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269830);