#6445781

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
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	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3044 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=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, 6352977);