#6450176

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
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
prime radical	#62845	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3022 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:

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

Polycyclic presentation:

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

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, 6357372);