#509585

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	#3155	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

476 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)(11,18)(12,20)(13,29)(14,31)(15,21)(16,33)(17,22)(19,26)(23,38)(24,36)(25,35)(27,40)(28,39)(30,34)(32,37);(1,3,5,7)(2,6,8,4)(9,11,17,13)(10,18,22,29)(12,25,28,32)(14,23,27,26)(15,30,16,24)(19,31,38,40)(20,35,39,37)(21,34,33,36);(1,4,5,6)(2,7,8,3)(9,12)(11,23)(13,26)(14,16)(15,27)(17,28)(18,34)(20,31)(21,33)(24,32)(25,30)(29,36)(35,37)(39,40);(2,8)(3,7)(9,13,17,11)(10,19,22,38)(12,26,28,23)(14,32,27,25)(15,30,16,24)(18,31,29,40)(20,36,39,34)(21,37,33,35)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=e,d^2=i,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, 416781);