#498798

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405994);