Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12941 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4,6,5)(8,14)(9,19)(10,11)(12,24)(15,31)(16,17)(18,30)(20,21)(22,38)(23,27)(25,26)(28,37)(29,33)(34,35);(3,5,6,4)(7,8)(9,15)(10,23)(11,14)(12,27)(13,18)(16,25)(17,24)(19,32)(20,29)(21,31)(22,35)(26,30)(28,34)(33,37)(36,38);(7,9,13,22)(8,15,18,35)(10,19,25,38)(11,20,26,37)(12,28,24,21)(14,29,30,33)(16,36,23,32)(17,31,27,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401494);