Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60932 |
| 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 | |
6456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(4,7)(5,9)(11,12)(13,22)(14,26)(15,21)(16,20)(17,31)(18,25)(19,28)(23,39)(24,35)(27,42)(29,38)(30,34)(32,37)(33,40)(36,41);(3,4)(5,8)(6,7)(9,10)(11,13)(12,18)(14,27)(15,30)(16,25)(17,24)(19,36)(20,22)(21,35)(23,29)(26,39)(28,40)(31,34)(32,41)(33,37)(38,42);(4,7)(8,10)(11,14)(12,19)(13,23)(15,28)(16,29)(17,32)(18,33)(20,37)(21,38)(22,36)(24,40)(25,27)(26,31)(30,41)(34,42)(35,39);(3,5,6,9)(4,8,7,10)(12,20)(18,22)(19,37)(21,31)(26,38)(33,36)(34,35)(39,42);(11,15,16,17)(12,21,20,31)(13,24,25,30)(14,28,29,32)(18,34,22,35)(19,38,37,26)(23,40,27,41)(33,42,36,39)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482837);