Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#10900 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #386 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1966 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);(3,4,6,5)(7,8)(9,18)(10,23)(11,27)(12,16)(13,17)(14,20)(15,25)(19,34)(21,30)(22,29)(24,32)(26,31)(28,36)(33,35)(37,38);(7,9,12,20,13,22,11,21)(8,14,16,29,17,30,27,18)(10,19,25,35,26,36,24,37)(15,28,31,38,32,34,23,33);(3,5,6,4)(7,10)(8,15)(9,19)(11,24)(12,25)(13,26)(14,28)(16,31)(17,32)(18,33)(20,35)(21,37)(22,36)(23,27)(29,38)(30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261125);