#6466562

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1908 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)(3,7)(4,5)(6,8)(9,10)(11,22)(12,29)(13,18)(14,19)(15,21)(16,26)(17,31)(20,23)(24,39)(25,38)(27,35)(28,36)(30,33)(32,37)(34,40);(2,5)(6,7)(9,11,23,28,15,27,14,26)(10,16,19,35,21,36,20,22)(12,30,32,40,31,25,13,24)(17,34,37,33,29,39,18,38);(1,3,4,8)(2,6,5,7)(9,12,23,32,15,31,14,13)(10,17,20,18,21,29,19,37)(11,24,26,25,27,40,28,30)(16,33,35,39,36,38,22,34);(9,13,14,31,15,32,23,12)(10,18,19,17,21,37,20,29)(11,25,28,24,27,30,26,40)(16,34,22,38,36,39,35,33);(9,14,15,23)(10,19,21,20)(11,26,27,28)(12,13,31,32)(16,35,36,22)(17,37,29,18)(24,25,40,30)(33,39,38,34)

Polycyclic presentation:

2^9|b^2=fg,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,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, 6373758);