Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60754 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
8018 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)(5,9)(6,8)(7,10)(11,12)(13,17)(14,24)(15,27)(16,19)(18,25)(20,29)(21,35)(22,31)(23,39)(26,34)(28,33)(30,37)(32,38)(36,42)(40,41);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,21)(15,23)(16,22)(18,30)(19,31)(20,32)(24,35)(25,37)(26,36)(27,39)(28,40)(29,38)(33,41)(34,42);(5,10)(7,9)(11,14,16,26)(12,18,19,33)(13,21,22,36)(15,28,29,25)(17,30,31,41)(20,34,27,24)(23,40,38,37)(32,42,39,35);(4,8)(7,9)(13,22)(17,31)(21,36)(23,38)(30,41)(32,39)(35,42)(37,40);(12,19)(17,31)(18,33)(20,27)(24,34)(30,41)(32,39)(35,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482614);