#8066937

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

4863 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)(9,10,13,12)(11,15,16,14)(17,18,23,25)(19,26,21,24)(20,29,30,22)(27,28,31,32);(1,3,6,4)(2,7,5,8)(9,11,13,16)(10,14,12,15)(17,19,23,21)(18,24,25,26)(20,27,30,31)(22,28,29,32);(2,5)(3,4)(19,21)(24,26)(27,31)(28,32);(2,5)(7,8)(9,12,13,10)(11,15,16,14)(17,20,23,30)(18,22,25,29)(19,27,21,31)(24,28,26,32);(1,4,6,3)(2,8,5,7)(17,21,23,19)(18,24,25,26)(20,27,30,31)(22,32,29,28);(1,5,6,2)(3,8,4,7)(17,22)(18,20)(19,28)(21,32)(23,29)(24,27)(25,30)(26,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=i,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, 7974133);