#6356919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
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	#61987	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264115);