Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(4,7)(17,18,21,19)(20,23,24,22);(2,6)(3,8)(9,10)(11,14)(12,15)(13,16)(18,19)(22,23);(1,2,5,6)(3,7,8,4)(9,11,13,15)(10,12,16,14)(17,19,21,18)(20,23,24,22);(3,8)(4,7)(9,11,13,15)(10,14,16,12)(17,20,21,24)(18,22,19,23);(1,3,5,8)(2,7,6,4)(9,10)(11,12)(13,16)(14,15);(1,4,5,7)(2,3,6,8)(9,12)(10,15)(11,16)(13,14)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9133401);