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) = C25 |
| direct product | C2×#7019 |
| 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 | |
726 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);(3,4)(5,7)(6,9)(8,10)(11,12,13,15,14,16,17,18)(19,20)(21,27,28,24)(22,26)(23,25)(29,33,34,31)(30,32);(4,7,9,8)(19,21,22,29,23,28,30,34)(20,24,26,31,25,27,32,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2226);