#6351505

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351503
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

(2,5)(3,7)(6,8)(9,10)(11,19)(12,16)(13,18)(14,17)(15,20)(21,23)(22,24);(3,7)(5,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,19)(17,24)(18,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19);(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,24,19);(10,17)(15,24)(16,18)(19,23)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258701);