#499168

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂²×#508	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1080 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)(11,12)(13,20)(14,22)(15,21)(16,33)(17,23)(18,19)(24,27)(25,28)(26,30)(29,34)(31,38)(32,39)(35,40)(36,37);(5,6)(7,8)(9,11)(10,18)(12,22)(13,24)(14,25)(15,19)(16,34)(17,26)(20,28)(21,29)(23,36)(27,38)(30,39)(31,37)(32,40)(33,35);(2,4)(6,8)(9,12,14,28,13,27,31,36,17,30,32,35,16,29,15,18)(10,19,21,34,33,40,39,26,23,37,38,24,20,25,22,11);(9,13,17,16)(10,20,23,33)(11,24,26,34)(12,27,30,29)(14,31,32,15)(18,28,36,35)(19,25,37,40)(21,22,38,39)

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=ghi,h^b=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406364);