#9054410

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

4556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8961606);