Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#7878 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C4⋊C4 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1622 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)(5,6)(7,8)(9,17)(10,14)(11,15)(12,16)(13,18)(19,22)(20,21)(23,24,26,29,27,30,33,36)(25,28,32,35,31,34,37,38);(4,6)(7,9,11,19,12,20,10,18)(8,13,14,21,16,22,15,17)(23,25)(24,28)(26,32)(27,31)(29,35)(30,34)(33,37)(36,38);(7,10,12,11)(8,14,16,15)(9,18,20,19)(13,21,22,17)(25,31)(28,34)(32,37)(35,38)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88116);