#6438653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3315 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)(3,4)(6,7)(9,10)(11,15)(12,14)(13,16)(17,24)(18,21)(19,22)(20,23);(5,6,8,7)(10,14)(11,17)(13,19)(15,23)(18,20)(21,24);(2,4)(5,7,8,6)(9,11,12,18)(10,15,14,21)(13,20,19,17)(16,23,22,24);(9,11,12,18)(10,15,14,21)(13,20,19,17)(16,23,22,24);(10,14)(15,21)(16,22)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345849);