#502127

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	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

767 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)(9,10,12,18,28,35,31,39,15,17,13,19,14,20,29,37)(11,21,23,40,32,38,30,33,26,22,24,34,25,36,27,16);(2,4)(5,6)(7,8)(9,10,12,18,28,35,31,39,15,17,13,19,14,20,29,37)(11,22,23,34,32,36,30,16,26,21,24,40,25,38,27,33);(6,8)(9,11,15,26)(10,16,17,33)(12,27,13,30)(14,32,28,25)(18,36,19,38)(20,40,35,34)(21,39,22,37)(23,31,24,29);(2,4)(10,17)(16,33)(18,19)(20,35)(21,22)(34,40)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,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, 409323);