Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51814 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(4,7)(8,10)(11,12,16,19)(13,14,17,21)(15,18,23,25)(20,22,24,26)(28,29)(31,39)(33,35)(37,42);(13,17)(14,21)(15,23)(18,25)(27,28,32,29)(30,35,38,33)(31,34,39,36)(37,41,42,40);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,20)(16,17)(18,22)(19,21)(23,24)(25,26)(27,29,32,28)(30,35,38,33)(31,36,39,34)(37,41,42,40);(3,5,6,9)(4,8,7,10)(11,14)(12,17)(13,19)(15,22)(16,21)(18,24)(20,25)(23,26)(27,30)(28,33)(29,35)(31,37)(32,38)(34,40)(36,41)(39,42);(11,15,16,23)(12,18,19,25)(13,20,17,24)(14,22,21,26)(27,31)(28,34)(29,36)(30,37)(32,39)(33,40)(35,41)(38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614914);