#6347222

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347203
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
prime radical	C₂×#3558	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3041 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)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,22,25)(19,27,29,32)(20,26,21,24)(23,30,31,28);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,31)(26,32);(1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16)(20,21)(24,26)(27,32)(28,30);(3,10)(7,14)(11,16)(13,15)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,27,31,32);(2,8)(6,9)(7,14)(13,15)(18,25)(23,31)(24,26)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254418);