#6349288

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
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

2965 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6256484);