#6353060

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352941
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
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3177 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=hi,e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260256);