#505186

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
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	#33522	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

663 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,14)(13,31)(15,20)(16,21)(17,25)(18,32)(19,29)(23,38)(24,37)(26,36)(27,28)(30,33)(34,39)(35,40);(2,5)(6,8)(9,11,14,25,16,27,29,26)(10,17,19,22,21,36,12,28)(13,24,32,38,33,40,15,34)(18,35,31,23,20,37,30,39);(1,3,4,7)(2,6,5,8)(9,12,29,21,16,19,14,10)(11,23,26,35,27,39,25,37)(13,31,15,18,33,30,32,20)(17,34,28,40,36,38,22,24);(2,5)(6,8)(9,13,14,32,16,33,29,15)(10,18,19,31,21,20,12,30)(11,24,25,38,27,40,26,34)(17,35,22,23,36,37,28,39)

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=f,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=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, 412382);