Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61275 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3493 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6452 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);(3,4,7,6)(5,9,10,8)(11,12)(13,22)(14,26)(15,20)(16,21)(17,30)(18,25)(19,27)(23,38)(24,34)(28,35)(29,33)(31,36)(32,39)(37,41)(40,42);(3,5)(4,8)(6,9)(7,10)(11,13)(12,18)(14,23)(15,29)(16,25)(17,24)(19,32)(20,34)(21,22)(26,42)(27,41)(28,40)(30,33)(31,39)(35,38)(36,37);(3,6,7,4)(5,9,10,8)(11,14,16,28)(12,19,21,36)(13,23,25,40)(15,27,17,31)(18,32,22,37)(20,35,30,26)(24,39,29,41)(33,42,34,38);(5,10)(8,9)(11,15,16,17)(12,20,21,30)(13,24,25,29)(14,27,28,31)(18,33,22,34)(19,35,36,26)(23,39,40,41)(32,42,37,38);(12,21)(13,25)(19,36)(20,30)(23,40)(24,29)(26,35)(39,41)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483815);