#502130

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

795 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)(9,10,17,32,13,28,31,40,16,22,14,21,15,20,29,34)(11,18,27,38,26,35,33,39,12,25,24,36,23,37,30,19);(2,4)(5,6)(7,8)(9,11,14,24,13,26,29,30,16,12,17,27,15,23,31,33)(10,18,21,36,28,35,34,19,22,25,32,38,20,37,40,39);(6,8)(9,12)(10,19)(11,16)(13,26)(14,30)(15,23)(17,33)(18,34)(20,38)(21,35)(22,39)(24,29)(25,40)(27,31)(28,36)(32,37);(2,4)(9,13,16,15)(10,20,22,28)(11,23,12,26)(14,29,17,31)(18,35,25,37)(19,38,39,36)(21,40,32,34)(24,33,27,30)

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=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, 409326);