#6349525

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349437
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

3001 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)(11,21)(12,23)(13,20)(14,18)(15,19)(16,22)(17,24);(1,2,5,8)(3,6,4,7)(9,11,15,17)(10,16,19,12)(13,22,14,23)(18,21,20,24);(1,3,5,4)(2,6,8,7)(9,12,15,16)(10,17,19,11)(13,24,14,21)(18,23,20,22);(9,13,15,14)(10,18,19,20)(11,22,17,23)(12,24,16,21);(1,3,5,4)(2,7,8,6)(10,19)(11,17)(18,20)(22,23)

Polycyclic presentation:

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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256721);