#6353917

order	512
nilpotency class	3
isoclinism	isoclinic class of #19254
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1764 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,26)(20,25)(21,29)(22,24)(23,27)(28,32)(30,31);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16);(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(18,24)(23,31)(25,29)(26,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,22,21)(18,25,24,29)(19,27,28,30)(23,32,31,26)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321137, #72321187, #72321231, …
maximal normal subgroup of	#72321137, #72321187, #72321231, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261113);