#504787

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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

551 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)(9,10,14,20,34,12,26,25,17,23,15,21,16,22,18,11)(13,24,28,37,30,40,35,38,32,36,29,27,31,39,33,19);(2,5)(6,7)(9,11)(10,18)(12,15)(13,24)(14,22)(16,20)(17,25)(19,28)(21,34)(23,26)(27,35)(29,38)(30,39)(31,40)(32,36)(33,37);(9,12,17,22)(10,16,23,34)(11,18,25,26)(13,27,32,37)(14,20,15,21)(19,30,38,31)(24,28,36,29)(33,39,35,40);(1,3,4,8)(2,6,5,7)(9,13,17,32)(10,19,23,38)(11,24,25,36)(12,27,22,37)(14,33,15,35)(16,30,34,31)(18,28,26,29)(20,39,21,40)

Polycyclic presentation:

2^9|a^2=e,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, 411983);