Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60927 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3062 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6472 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);(4,7)(5,9)(11,12)(13,21)(14,25)(15,19)(16,30)(17,23)(18,28)(20,29)(22,38)(24,33)(26,42)(27,36)(31,35)(32,41)(34,39)(37,40);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,26)(15,23)(16,24)(18,34)(19,21)(20,33)(22,27)(25,38)(28,41)(29,40)(30,37)(31,39)(32,35)(36,42);(4,7)(8,10)(11,14,15,27)(12,18,19,35)(13,22,23,26)(16,31,29,28)(17,32,21,34)(20,36,30,25)(24,41,40,39)(33,38,37,42);(3,5,6,9)(4,8,7,10)(12,19)(17,21)(18,35)(20,30)(25,36)(32,34)(33,37)(38,42);(13,23)(17,21)(22,26)(24,40)(32,34)(33,37)(38,42)(39,41)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482830);