#6445793

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2941 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352989);