Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417460 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18,21,25,23,20,22,30)(19,26,28,27,29,32,31,24);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,13,14,16);(1,2,5,4)(3,6,8,7)(11,15)(13,16)(17,19)(18,24)(20,27)(21,31)(22,28)(23,29)(25,32)(26,30);(1,4,5,2)(3,6,8,7)(17,20,23,18)(19,27,29,24)(21,30,22,25)(26,28,32,31)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=e,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348390);