#502482

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	#28225	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

727 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,d^2=e,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, 409678);