Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12727 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12730 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#519 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,8,13,9,14,5,15)(6,18,21,29,20,11,16,27)(7,12,17,25,23,28,22,10)(19,32,30,24,34,26,33,31);(3,5,9,8)(4,10,14,25)(6,16,20,21)(7,22,23,17)(11,24,18,31)(12,13,28,15)(19,33,34,30)(26,27,32,29);(3,6,9,20)(4,11)(5,16,8,21)(7,19,23,34)(10,24)(12,26)(13,27)(14,18)(15,29)(17,30,22,33)(25,31)(28,32)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400664);