#501065

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,h^c=hi

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, 408261);