Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12820 |
| 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 | |
1114 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)(9,17)(10,22)(11,26)(12,28)(13,18)(14,20)(15,19)(16,27)(21,35)(23,34)(24,38)(25,30)(29,31)(32,36)(33,37);(7,9,11,20,13,21,27,12)(8,14,16,17,18,28,26,35)(10,19,23,36,25,37,24,29)(15,30,31,34,33,22,32,38);(7,10)(8,15)(9,19)(11,23)(12,29)(13,25)(14,30)(16,31)(17,34)(18,33)(20,36)(21,37)(22,28)(24,27)(26,32)(35,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400976);