#6501556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499780
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23962	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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)(3,7)(4,8)(5,6)(17,18)(19,23)(20,27)(21,30)(22,25)(24,26)(28,32)(29,31);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,29)(18,24,25,32)(19,26,21,28)(23,31,30,27);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(20,29)(24,32)(26,28)(27,31);(10,14)(13,15)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,31,32,27)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,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, 6408752);