#6446131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=fg,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6353327);