Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#55816 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4734 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)(5,9)(11,12)(13,19)(14,21)(15,23)(16,17)(18,24)(20,22)(25,26)(27,28,33,31)(29,36,30,34)(32,39,40,35)(37,41,38,42);(4,7)(8,10)(12,17)(14,19)(15,24)(20,26)(27,29,33,30)(28,34,31,36)(32,37,40,38)(35,41,39,42);(3,4)(5,8)(6,7)(9,10)(11,13,16,21)(12,14,17,19)(15,20,24,26)(18,22,23,25)(27,30,33,29)(28,34,31,36)(32,38,40,37)(35,41,39,42);(3,5,6,9)(4,8,7,10)(11,14)(12,13)(15,22)(16,19)(17,21)(18,20)(23,26)(24,25)(27,31,33,28)(29,34,30,36)(32,39,40,35)(37,41,38,42);(11,15)(12,18)(13,20)(14,22)(16,24)(17,23)(19,25)(21,26)(27,32)(28,35)(29,37)(30,38)(31,39)(33,40)(34,41)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009937);