#6350269

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3001 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339705, #72339708, #72339796, …
maximal normal subgroup of	#72339705, #72339708, #72339796, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257465);