#6351730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351720
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2923 conjugacy classes of subgroups; only the 566 normal subgroups are shown.

normal subgroups only

This group has 2923 conjugacy classes of subgroups — too many to draw. Only its 566 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235012, #72268538
maximal normal subgroup of	#72235012

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258926);