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×#57053 |
| 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);(3,4)(5,6)(7,8)(9,10)(11,12,16,13)(14,17,21,19)(15,20,23,18)(22,25,26,24)(28,32)(29,35)(30,38)(40,42);(4,8)(5,9)(12,13)(14,21)(15,23)(24,25)(27,28,31,32)(29,33,35,36)(30,39,38,34)(37,42,41,40);(11,13,16,12)(14,17,21,19)(15,20,23,18)(22,24,26,25)(27,29)(28,33)(30,37)(31,35)(32,36)(34,40)(38,41)(39,42);(3,5,7,9)(4,6,8,10)(11,14,16,21)(12,17,13,19)(15,22,23,26)(18,24,20,25)(27,28)(29,36)(30,34)(31,32)(33,35)(37,42)(38,39)(40,41);(3,6)(4,9)(5,8)(7,10)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,30,31,38)(28,34,32,39)(29,37,35,41)(33,40,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007349);