#6355300

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355299
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2157 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262496);