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) = C25×C4 |
| direct product | C2×#12614 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C8×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1122 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);(3,4)(5,6)(7,8,11,17,12,16,24,31,13,18,25,32,10,14,23,27)(9,15,20,29,21,28,34,37,22,30,35,38,19,26,33,36);(4,6)(8,14,18,16)(15,26,30,28)(17,27,32,31)(29,36,38,37);(7,9,11,20,12,21,24,34,13,22,25,35,10,19,23,33)(8,15,17,29,16,28,31,37,18,30,32,38,14,26,27,36)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400167);