#6466082

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
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

1930 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)(11,23)(12,18)(13,19)(14,20)(15,21)(16,22)(17,27)(24,35)(25,32)(26,34)(28,38)(29,40)(30,36)(31,37)(33,39);(9,11)(10,17)(12,29)(13,24)(14,26)(15,27)(16,28)(18,35)(19,32)(20,23)(21,38)(22,34)(25,31)(30,39)(33,37)(36,40);(1,3,4,8)(2,5,6,7)(9,12,15,13,16,31,14,30)(10,18,21,19,22,37,20,36)(11,24,26,29,28,39,27,25)(17,32,23,35,34,40,38,33);(9,13,15,31,16,30,14,12)(10,19,21,37,22,36,20,18)(11,25,26,24,28,29,27,39)(17,33,23,32,34,35,38,40);(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,30,31,13)(17,23,34,38)(18,36,37,19)(24,29,39,25)(32,35,40,33)

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,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, 6373278);