Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #481 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
790 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)(9,17,20,33)(10,13)(11,23)(12,26)(14,19,28,35)(15,25)(16,18)(21,36,27,31)(22,37)(24,30,38,32)(29,34);(3,4,6,5)(8,13)(11,22)(14,28)(16,26)(19,35)(24,38)(25,34)(30,32);(3,5,6,4)(7,9,18,21,10,20,12,27)(8,14,26,30,13,28,16,32)(11,24,34,35,22,38,25,19)(15,31,37,33,29,36,23,17)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402073);