#8066945

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7974141);