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×#60819 |
| 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 | |
7642 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)(13,22)(14,26)(15,20)(16,18)(17,23)(19,27)(21,29)(24,31)(25,30)(28,34)(32,37)(33,39)(35,36)(38,41)(40,42);(3,5,6,10)(4,7,8,9)(11,13)(12,17)(14,24)(15,23)(16,25)(18,32)(19,31)(20,22)(21,30)(26,35)(27,36)(28,38)(29,37)(33,41)(34,40)(39,42);(5,10)(7,9)(12,18)(13,23)(16,29)(17,30)(19,33)(20,21)(22,32)(24,36)(26,34)(28,39)(31,40)(35,41);(4,8)(7,9)(11,14)(12,19)(13,24)(15,27)(16,28)(17,31)(18,33)(20,26)(21,34)(22,35)(23,36)(25,38)(29,39)(30,40)(32,41)(37,42);(14,27)(19,26)(24,36)(28,39)(31,35)(33,34)(38,42)(40,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482723);