Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#60823 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7626 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)(13,23)(14,29)(15,21)(16,22)(17,33)(18,27)(19,30)(20,31)(24,39)(25,35)(26,40)(28,36)(32,38)(34,42)(37,41);(3,5,6,10)(4,7,8,9)(11,13)(12,18)(14,25)(15,26)(16,27)(17,28)(19,37)(20,35)(21,36)(22,23)(24,31)(29,39)(30,42)(32,41)(33,40)(34,38);(5,10)(7,9)(11,14)(12,19)(13,24)(15,30)(16,31)(17,32)(18,34)(20,21)(22,38)(23,37)(25,27)(26,41)(28,42)(29,33)(35,40)(36,39);(4,8)(7,9)(11,14)(12,20)(13,25)(15,32)(16,31)(17,30)(18,35)(19,21)(22,29)(23,39)(24,27)(26,41)(28,42)(33,38)(34,40)(36,37);(11,15,16,17)(12,21,22,33)(13,26,27,28)(14,30,31,32)(18,36,23,40)(19,20,38,29)(24,41,25,42)(34,39,37,35)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482726);