#499324

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 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)(24,28)(25,29)(26,34)(27,30)(31,39)(32,40)(35,37)(36,38);(5,6)(7,8)(9,11,15,25,16,26,32,38,17,27,31,37,13,24,14,19)(10,18,22,28,20,35,39,30,23,36,40,34,33,29,21,12);(2,4)(6,8)(9,12)(10,19)(11,21)(13,28)(14,18)(15,29)(16,34)(17,30)(20,37)(22,24)(23,38)(25,33)(26,40)(27,39)(31,35)(32,36);(9,13,17,16)(10,20,23,33)(11,24,27,26)(12,28,30,34)(14,31,32,15)(18,35,36,29)(19,37,38,25)(21,22,39,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,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^c=eh,f^c=fhi,g^c=ghi,h^c=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, 406520);