#6348014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347957
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	#62830	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2863 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255210);