#6360693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360376
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

1570 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6267889);