#6357402

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357390
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

2123 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264598);