Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60586 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7548 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);(3,4)(5,8)(6,7)(9,10)(11,12)(13,18)(14,20)(15,21)(16,30)(17,22)(19,27)(23,32)(24,33)(25,40)(26,34)(28,35)(29,36)(31,37)(38,41)(39,42);(4,7)(8,10)(11,13,17,26)(12,18,22,34)(14,23,28,38)(15,24,16,25)(19,31,36,42)(20,32,35,41)(21,33,30,40)(27,37,29,39);(11,14)(12,19)(13,23)(15,27)(16,29)(17,28)(18,31)(20,30)(21,35)(22,36)(24,37)(25,39)(26,38)(32,40)(33,41)(34,42);(3,5,6,9)(4,8,7,10)(11,14)(12,20)(13,23)(15,29)(16,27)(17,28)(18,32)(19,30)(21,36)(22,35)(24,39)(25,37)(26,38)(31,40)(33,42)(34,41);(11,15,17,16)(12,21,22,30)(13,24,26,25)(14,27,28,29)(18,33,34,40)(19,35,36,20)(23,37,38,39)(31,41,42,32)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482316);