Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#31072 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8)(9,17,21,28)(10,22,25,31)(11,16)(12,15)(13,20,29,19)(14,24,30,23)(18,34)(26,32)(27,37)(33,35)(36,38);(3,4)(5,6)(7,9)(8,13)(10,18)(11,19)(12,21)(14,27)(15,29)(16,28)(17,32)(20,26)(22,33)(23,35)(24,36)(25,37)(30,34)(31,38);(4,6)(7,10)(8,14)(9,18)(11,23)(12,25)(13,27)(15,30)(16,31)(17,33)(19,35)(20,36)(21,37)(22,32)(24,26)(28,38)(29,34);(8,15)(9,19)(11,26)(13,17)(14,30)(18,35)(20,21)(23,24)(27,33)(28,29)(34,38)(36,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410556);