#6355319

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

2196 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)(11,19)(12,13)(15,22)(17,18)(20,21)(23,24);(1,2,6,8,4,7,3,5)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24);(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,22)(18,24);(10,16)(15,23)(17,18)(22,24);(9,12,14,13)(10,17,16,18)(11,19,20,21)(15,22,23,24)

Polycyclic presentation:

2^9|b^2=fh,e^2=i,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, 6262515);