#6350917

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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

3161 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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,16)(14,15)(17,18)(19,27)(20,23)(21,24)(22,26)(25,28)(29,32)(30,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(18,23)(20,21)(24,26)(25,31)(27,32)(29,30);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,15)(11,16)(18,24)(19,28)(20,21)(23,26)(25,31)(27,32);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,8,15,16)(17,19,22,28)(18,25,26,27)(20,29,21,30)(23,31,24,32);(19,28)(25,27)(29,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258113);