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×#12945 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| 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 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1082 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)(7,8)(9,20)(10,14)(11,29)(12,30)(13,19)(15,22)(16,25)(17,28)(18,27)(21,34)(23,33)(24,36)(26,35)(31,32)(37,38);(3,5,6,4)(8,14)(10,25)(11,28)(12,27)(15,32)(16,18)(17,19)(20,34)(21,22)(23,31)(26,38)(29,30)(33,35)(36,37);(7,9)(8,15)(10,26)(11,21)(12,31)(13,24)(14,32)(16,36)(17,33)(18,37)(19,35)(20,29)(22,28)(23,27)(25,38)(30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401507);