#6578288

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

1926 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fi,e^2=f,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, 6485484);