#504786

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	#13325	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,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, 411982);