#6454813

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3151 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)(3,6)(4,5)(7,8)(9,10,16,20)(11,14,19,15)(12,21,23,17)(13,18,22,24);(9,11,15,10,16,19,14,20)(12,17,22,24,23,21,13,18);(2,5)(6,8)(9,12,16,23)(10,17,20,21)(11,18,19,24)(13,15,22,14);(9,13)(10,18)(11,21)(12,15)(14,23)(16,22)(17,19)(20,24);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,20,11)(12,13,23,22)(17,24,21,18)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^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, 6362009);