#6491477

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2987 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398673);