#6349523

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

Polycyclic presentation:

2^9|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, 6256719);