Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#33410 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1946 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)(10,14)(11,21)(15,23)(16,31)(17,26)(19,27)(20,33)(22,28)(24,25)(29,34)(30,38)(32,37)(35,36);(7,8,14,25,11,16,22,26,12,17,28,31,21,24,10,13)(9,15,27,36,20,30,32,34,18,29,37,38,33,35,19,23);(7,9)(8,15)(10,19)(11,20)(12,18)(13,23)(14,27)(16,30)(17,29)(21,33)(22,32)(24,35)(25,36)(26,34)(28,37)(31,38);(9,18)(15,29)(19,32)(20,33)(23,34)(27,37)(30,35)(36,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529730);