Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13122 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C8×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
808 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);(4,5)(7,8,13,18,14,17,32,35,11,16,20,21,12,29,31,37)(9,19,24,36,25,30,34,38,22,10,26,28,23,27,15,33);(3,4,6,5)(7,9,20,26,14,25,31,15,11,22,13,24,12,23,32,34)(8,10,21,36,17,27,37,38,16,19,18,28,29,30,35,33);(3,5,6,4)(7,10,11,19)(8,15,16,34)(9,21,22,18)(12,30,14,27)(13,28,20,36)(17,24,29,26)(23,35,25,37)(31,38,32,33)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402024);