#509692

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13319	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

557 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)(9,10,14,17)(11,21,24,15)(12,22,27,40)(13,29,30,23)(16,19,33,20)(18,38,25,36)(26,37,35,28)(31,39,32,34);(2,5)(3,8)(9,11)(10,15)(12,26)(13,29)(14,24)(16,25)(17,21)(18,33)(19,38)(20,36)(22,28)(23,30)(27,35)(31,34)(32,39)(37,40);(9,12)(10,13)(11,22)(14,27)(15,34)(16,37)(17,30)(18,35)(19,32)(20,31)(21,39)(23,36)(24,40)(25,26)(28,33)(29,38);(1,3,4,8)(2,6,5,7)(9,13)(10,12)(11,23)(14,30)(15,35)(16,31)(17,27)(18,34)(19,28)(20,37)(21,26)(22,36)(24,29)(25,39)(32,33)(38,40)

Polycyclic presentation:

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

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, 416888);