#6356993

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	#61977	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1549 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:

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

Polycyclic presentation:

2^9|b^2=fi,e^2=i,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, 6264189);