#6357523

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2248 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=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, 6264719);