#6354168

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1666 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261364);