#505184

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	C₂×#576	C₂²
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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,24)(13,23)(14,20)(16,19)(17,21)(18,28)(22,30)(25,26)(27,36)(29,37)(31,33)(32,38)(34,40)(35,39);(2,5)(6,8)(9,11,15,10,17,20,14,21)(12,25,28,35,30,37,27,40)(13,23,31,19,32,38,16,33)(18,26,24,34,36,29,22,39);(1,3,4,7)(2,6,5,8)(9,12,14,27,17,30,15,28)(10,18,11,22,21,36,20,24)(13,29,16,34,32,26,31,39)(19,37,23,35,33,25,38,40);(2,5)(6,8)(9,13,15,31,17,32,14,16)(10,19,20,38,21,33,11,23)(12,26,28,34,30,29,27,39)(18,25,24,35,36,37,22,40)

Polycyclic presentation:

2^9|b^2=f,c^2=e,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, 412380);