#6357508

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357480
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

1963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18)(19,27)(20,22)(21,25)(23,26)(24,30)(28,31)(29,32);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,22,30,23,31,21,29)(18,24,25,27,26,32,20,28);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,12,13,16)(17,20,21,26,23,25,22,18)(19,28,29,32,31,27,30,24);(2,8)(6,13)(7,14)(12,16);(3,9)(7,14)(10,15)(12,16)(17,21,23,22)(18,25,26,20)(19,29,31,30)(24,27,32,28)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,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, 6264704);