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×#50997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4486 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);(4,8)(9,10)(11,12,16,19)(13,17,20,25)(14,15,18,24)(21,26,23,22)(27,28,33,35)(29,36,30,31)(32,40,41,34)(37,39,38,42);(13,20)(14,18)(15,24)(17,25)(27,29,33,30)(28,31,35,36)(32,37,41,38)(34,39,40,42);(3,4)(5,9)(6,10)(7,8)(11,13)(12,17)(14,23)(15,22)(16,20)(18,21)(19,25)(24,26)(27,30,33,29)(28,31,35,36)(32,38,41,37)(34,39,40,42);(3,5,7,6)(4,9,8,10)(11,14,16,18)(12,15,19,24)(13,21,20,23)(17,26,25,22)(27,31)(28,29)(30,35)(32,39)(33,36)(34,37)(38,40)(41,42);(3,6,7,5)(4,10,8,9)(11,15)(12,18)(13,22)(14,19)(16,24)(17,21)(20,26)(23,25)(27,32)(28,34)(29,37)(30,38)(31,39)(33,41)(35,40)(36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613843);