Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #481 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1410 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);(4,6)(7,8,11,17,28,30,25,16,13,15,26,34,12,18,10,24)(9,19,22,14,29,32,20,36,21,37,38,31,23,35,27,33);(3,4,5,6)(7,9,12,29,13,21,28,23)(8,14,30,33,15,31,18,36)(10,22,26,20,25,38,11,27)(16,35,34,37,24,32,17,19);(8,15)(10,25)(11,26)(14,31)(18,30)(20,27)(22,38)(33,36)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401679);