#6576090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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

1856 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483286);