#6363169

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363086
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1564 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6270365);