Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#12664 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C8×D4 |
| prime radical | C24 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,12,17,13,18,30,27,14,19,31,26,11,16,25,10)(9,20,23,21,22,34,37,36,24,35,29,38,28,15,32,33);(3,4,5,6)(7,9)(8,15)(10,21)(11,28)(12,29)(13,22)(14,24)(16,35)(17,36)(18,20)(19,34)(23,31)(25,37)(26,33)(27,38)(30,32);(7,10,25,16,11,26,31,19,14,27,30,18,13,17,12,8)(9,21,37,35,28,33,23,34,24,38,32,20,22,36,29,15)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400317);