#502128

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

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=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, 409324);