Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8309 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2886 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)(9,15,17,12)(10,13)(11,14)(16,21,22,19)(18,20)(23,24,26,30,27,29,35,37)(25,31,33,38,32,28,34,36);(7,9)(8,12)(10,16)(11,17)(13,19)(14,15)(18,22)(20,21)(23,25)(24,28)(26,34)(27,32)(29,31)(30,38)(33,35)(36,37);(4,6)(7,10,11,18)(8,13,14,20)(9,16,17,22)(12,19,15,21)(25,32)(28,31)(33,34)(36,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91956);