#6347303

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347252
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3078 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,32)(29,31);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,16)(10,14)(17,19)(18,22)(20,29)(21,26)(23,31)(24,32)(25,28)(27,30);(20,30)(24,28)(25,32)(27,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(18,23)(22,31)(24,28)(25,32);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(19,26)(22,31)(25,32)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254499);