Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7113 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
620 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)(5,9,10,7)(6,8)(11,12,13,15,14,16,17,18)(20,24)(21,27)(22,29)(25,34)(26,33)(28,31)(30,32);(3,5)(4,7)(6,10)(8,9)(19,20,23,26)(21,28,30,34)(22,31,29,25)(24,32,33,27)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31264);