Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#7018 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #95 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(4,8)(5,6)(9,10)(11,12,13,15,14,16,17,18)(19,20)(21,27,29,33)(22,25)(23,26)(24,28,31,34)(30,32);(3,4,6,9,7,10,5,8)(19,21)(20,24,26,31)(22,28)(23,29)(25,27,32,33)(30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2190);