#6357809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357798
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

1966 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18)(19,21)(20,26)(22,27)(23,25)(24,28)(29,31)(30,32);(2,6)(7,14)(8,13)(9,15)(17,19,21,18,23,27,22,25)(20,26,30,24,29,31,28,32);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20,22,28,23,29,21,30)(18,24,19,26,25,32,27,31);(3,10)(7,14)(9,15)(11,16)(17,21,23,22)(18,19,25,27)(20,28,29,30)(24,31,32,26);(20,29)(24,32)(26,31)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265005);