#502112

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

798 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)(9,10,15,22,13,29,30,34,14,21,16,23,17,20,31,40)(11,24,26,37,28,38,32,19,12,18,27,36,25,35,33,39);(2,4)(5,6)(7,8)(9,11,15,26,13,28,30,32,14,12,16,27,17,25,31,33)(10,18,22,36,29,35,34,39,21,24,23,37,20,38,40,19);(6,8)(9,12,14,11)(10,19,21,39)(13,28,17,25)(15,32,16,33)(18,34,24,40)(20,36,29,37)(22,38,23,35)(26,31,27,30);(2,4)(9,13,14,17)(10,20,21,29)(11,25,12,28)(15,30,16,31)(18,35,24,38)(19,36,39,37)(22,40,23,34)(26,33,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,g^c=gh,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, 409308);