#509547

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=ei,d^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416743);