#491781

order	512
nilpotency class	4
isoclinism	isoclinic class of #491712
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	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

461 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287072, #72287259, #72309493, …
maximal normal subgroup of	#72287072, #72287259, #72309493, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398977);