#501071

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,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, 408267);