#502115

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	#24778	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)(9,10,13,18,27,31,28,26,15,20,12,17,14,19,30,29)(11,16,24,35,38,39,32,37,22,33,23,34,25,36,40,21);(2,4)(5,6)(7,8)(9,11,12,23,27,38,30,40,15,22,13,24,14,25,28,32)(10,16,17,34,31,39,29,21,20,33,18,35,19,36,26,37);(6,8)(9,10,15,20)(11,21,22,37)(12,26,13,29)(14,31,27,19)(16,32,33,40)(17,30,18,28)(23,39,24,36)(25,35,38,34);(2,4)(11,22)(16,33)(21,37)(23,24)(25,38)(32,40)(34,35)(36,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,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, 409311);