#6542224

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533063
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×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3016 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449420);