#6350381

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

2841 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)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(18,24)(19,26)(20,28)(21,22)(25,27)(31,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,18)(19,21)(20,25)(22,24)(23,26)(27,29)(28,32)(30,31);(17,19,23,24)(18,21,26,22)(20,27,28,31)(25,29,32,30);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(20,28)(25,32)(27,31)(29,30)

Polycyclic presentation:

2^9|c^2=i,e^2=hi,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	#72339709, #72339717, #72339796, …
maximal normal subgroup of	#72339709, #72339717, #72339796, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257577);