#6466306

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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)(11,22)(12,17)(13,18)(14,20)(15,21)(16,25)(19,29)(23,35)(24,32)(26,34)(27,38)(28,40)(30,36)(31,37)(33,39);(9,11,14,25,15,27,29,26)(10,16,19,22,21,34,20,38)(12,28,13,23,30,24,31,39)(17,35,37,40,36,33,18,32);(1,3,4,8)(2,5,6,7)(9,12,29,31,15,30,14,13)(10,17,19,37,21,36,20,18)(11,23,25,24,27,39,26,28)(16,32,38,33,34,40,22,35);(9,13,14,30,15,31,29,12)(10,18,20,36,21,37,19,17)(11,24,26,23,27,28,25,39)(16,33,22,32,34,35,38,40);(12,30)(13,31)(17,36)(18,37)(23,39)(24,28)(32,40)(33,35)

Polycyclic presentation:

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

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, 6373502);