Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1324 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)(7,8,13,20,33,34,32,18,15,22,12,19,14,21,11,31)(9,23,26,16,30,17,24,37,28,38,25,36,27,35,10,29);(7,9,14,30,15,28,33,27)(8,16,34,29,22,36,21,37)(10,11,26,12,24,32,25,13)(17,20,23,18,35,19,38,31);(4,6)(7,10)(8,17)(9,11)(12,30)(13,27)(14,26)(15,24)(16,20)(18,29)(19,36)(21,38)(22,35)(23,34)(25,33)(28,32)(31,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401736);