#501854

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	#33527	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

651 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,23)(16,20)(17,24)(18,22)(19,35)(21,33)(25,30)(26,31)(27,38)(28,37)(29,40)(32,36)(34,39);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,27)(13,23)(14,18)(15,25)(16,36)(19,32)(20,37)(21,22)(24,35)(26,33)(28,34)(29,31)(30,39)(38,40);(2,6)(5,8)(9,12,15,10)(11,22,25,37)(13,28,30,18)(14,29,34,19)(16,20,33,21)(17,36,27,26)(23,35,39,40)(24,31,38,32);(9,13)(10,18)(11,23)(12,28)(14,17)(15,30)(16,31)(19,26)(20,38)(21,24)(22,35)(25,39)(27,34)(29,36)(32,33)(37,40)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,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, 409050);