#6466078

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373274);