#6349201

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349137
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,f^c=fi,g^c=gi

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, 6256397);