#502106

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

800 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)(9,10,16,32,30,11,17,33,15,21,14,20,13,19,35,23)(12,24,39,22,26,40,31,18,29,38,27,36,28,25,34,37);(2,4)(5,6)(7,8)(9,11,14,23,30,21,35,32,15,19,16,33,13,10,17,20)(12,25,27,18,26,24,34,36,29,40,39,37,28,38,31,22);(6,8)(9,12)(10,18)(11,22)(13,26)(14,31)(15,29)(16,34)(17,27)(19,36)(20,25)(21,37)(23,38)(24,33)(28,30)(32,40)(35,39);(2,4)(9,13,15,30)(10,19,21,11)(12,26,29,28)(14,17,16,35)(18,36,37,22)(20,33,32,23)(24,40,38,25)(27,34,39,31)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409302);