#6354828

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354774
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1657 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)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18)(19,21)(20,28)(22,27)(23,25)(24,30)(26,31)(29,32);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19,21,18,23,27,22,25)(20,26,30,24,31,28,29,32);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,20)(18,24)(19,26)(21,30)(22,29)(23,31)(25,32)(27,28);(3,10)(8,15)(11,14)(12,16)(17,21,23,22)(18,19,25,27)(20,29,31,30)(24,28,32,26);(2,9)(6,13)(8,15)(12,16)(17,21,23,22)(18,19,25,27)(20,30,31,29)(24,26,32,28)

Polycyclic presentation:

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

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, 6262024);