#503528

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

784 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

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, 410724);