Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
| direct product | C2×C16×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C4×C8 | C24 |
| socle, prime radical | C23 | C8×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18);(19,20,21,22,23,24,25,26)(27,34,33,32,31,30,29,28);(19,27,23,31)(20,28,24,32)(21,29,25,33)(22,30,26,34)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390726);