#504793

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411989);