Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60519 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7946 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)(5,8)(6,7)(9,10)(11,12,17,22)(13,18,24,32)(14,27,29,36)(15,21,16,20)(19,30,35,28)(23,37,38,42)(25,34,26,33)(31,40,41,39);(4,7)(8,10)(11,13)(12,18)(14,23)(15,25)(16,26)(17,24)(19,31)(20,33)(21,34)(22,32)(27,37)(28,39)(29,38)(30,40)(35,41)(36,42);(3,5,6,9)(4,8,7,10)(11,14)(12,19)(13,23)(15,28)(16,30)(17,29)(18,31)(20,27)(21,36)(22,35)(24,38)(25,39)(26,40)(32,41)(33,37)(34,42);(13,24)(18,32)(23,38)(25,26)(31,41)(33,34)(37,42)(39,40);(11,15,17,16)(12,20,22,21)(13,25,24,26)(14,28,29,30)(18,33,32,34)(19,27,35,36)(23,39,38,40)(31,37,41,42)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482250);