Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#33382 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,5,6)(8,13)(10,20)(11,22)(14,26)(15,30)(16,27)(18,33)(19,32)(21,24)(23,25)(28,37)(29,38)(31,36)(34,35);(7,8,12,16)(9,14,17,28)(10,21,23,30)(11,24,22,15)(13,25,27,20)(18,34,31,38)(19,35,32,29)(26,36,37,33);(7,9)(8,14)(10,18)(11,19)(12,17)(13,26)(15,29)(16,28)(20,33)(21,34)(22,32)(23,31)(24,35)(25,36)(27,37)(30,38);(9,17)(14,28)(18,31)(19,32)(26,37)(29,35)(33,36)(34,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529691);