#502338

order	512
nilpotency class	4
isoclinism	isoclinic class of #502133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi,h^c=hi

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, 409534);