#502337

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	C₂×#2206	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 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)(9,10,12,18,26,37,29,36,15,20,27,35,14,19,13,17)(11,16,22,31,39,40,23,32,25,30,21,38,24,33,28,34);(2,5)(6,7)(9,11,12,22,26,39,29,23,15,25,27,21,14,24,13,28)(10,16,18,31,37,40,36,32,20,30,35,38,19,33,17,34);(10,17)(11,21)(12,13)(14,26)(16,30)(18,19)(20,36)(22,25)(23,39)(24,28)(27,29)(31,32)(34,38)(35,37);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=f,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, 409533);