Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47196 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(7,10)(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,32,34)(29,35,36,31)(30,33,40,42)(37,41,38,39);(3,4,6,8)(5,7,9,10)(12,16)(13,19)(14,23)(24,26)(29,36)(31,35)(37,38)(39,41);(3,5)(4,7)(6,9)(8,10)(11,13,15,19)(12,17,16,20)(14,21,23,25)(18,24,22,26)(27,29)(28,31)(30,38)(32,36)(33,41)(34,35)(37,40)(39,42);(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,30)(28,33)(29,37)(31,39)(32,40)(34,42)(35,41)(36,38);(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,25)(20,26)(27,31)(28,29)(30,39)(32,35)(33,37)(34,36)(38,42)(40,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608399);