Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#13115 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1919 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
590 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);(3,4,9,14,7,13,10,24)(5,15,18,32,17,11,8,22)(6,19,20,26,21,34,25,31)(12,27,28,33,30,23,29,16);(3,5,9,18,7,17,10,8)(4,11,14,22,13,15,24,32)(6,16,20,27,21,33,25,23)(12,26,28,34,30,31,29,19);(3,6,9,20,7,21,10,25)(4,12,24,29,13,30,14,28)(5,16,18,27,17,33,8,23)(11,26,32,19,15,31,22,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402004);