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×#51253 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4676 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)(13,16)(14,17)(15,20)(18,23)(19,22)(21,25)(24,26)(27,28,31,34)(29,32,37,35)(30,33,39,38)(36,40,42,41);(12,13)(15,23)(17,19)(21,26)(27,28,31,34)(29,35,37,32)(30,38,39,33)(36,40,42,41);(3,4)(5,8)(6,7)(9,10)(11,13,16,12)(14,17,22,19)(15,20,23,18)(21,24,26,25)(27,28)(29,35)(30,33)(31,34)(32,37)(36,41)(38,39)(40,42);(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25)(27,29,31,37)(28,32,34,35)(30,36,39,42)(33,40,38,41);(3,5,6,9)(4,8,7,10)(11,15)(12,18)(13,20)(14,21)(16,23)(17,24)(19,25)(22,26)(27,30,31,39)(28,33,34,38)(29,36,37,42)(32,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614185);