#504792

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

Polycyclic presentation:

2^9|a^2=ei,b^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, 411988);