#6574592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 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)(11,18)(12,26)(13,16)(14,19)(15,21)(17,29)(20,32)(22,39)(23,25)(24,31)(27,34)(28,37)(30,35)(33,36)(38,40);(1,3,4,8)(2,5,6,7)(9,11,14,13)(10,16,19,18)(12,27,28,40)(15,24,32,25)(17,35,36,39)(20,31,21,23)(22,33,30,29)(26,38,37,34);(9,12,14,28)(10,17,19,36)(11,22,13,30)(15,33,32,29)(16,34,18,38)(20,37,21,26)(23,35,31,39)(24,27,25,40);(9,13,14,11)(10,18,19,16)(12,27,28,40)(15,25,32,24)(17,35,36,39)(20,23,21,31)(22,33,30,29)(26,38,37,34);(9,10)(11,23)(12,17)(13,31)(14,19)(15,20)(16,24)(18,25)(21,32)(22,35)(26,29)(27,34)(28,36)(30,39)(33,37)(38,40)

Polycyclic presentation:

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

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, 6481788);