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×#56430 |
| 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 | |
4534 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);(11,12)(13,17)(14,18)(15,16)(19,21)(20,25)(22,24)(23,26)(27,28,32,35)(29,33,30,37)(31,34,39,42)(36,40,38,41);(3,4)(5,9)(6,8)(7,10)(12,15)(13,19)(14,22)(25,26)(27,29,32,30)(28,33,35,37)(31,38,39,36)(34,41,42,40);(3,4,6,8)(5,7,10,9)(11,13,16,19)(12,17,15,21)(14,23,22,20)(18,26,24,25)(27,28)(29,33)(30,37)(31,34)(32,35)(36,40)(38,41)(39,42);(3,4)(5,7)(6,8)(9,10)(11,14,16,22)(12,18,15,24)(13,20,19,23)(17,25,21,26)(27,30,32,29)(28,33,35,37)(31,38,39,36)(34,40,42,41);(3,5)(4,7)(6,10)(8,9)(11,15,16,12)(13,21,19,17)(14,24,22,18)(20,26,23,25)(27,31)(28,34)(29,36)(30,38)(32,39)(33,40)(35,42)(37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006333);