Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62861 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
33634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 42 points:
(1,2);(3,4,6,8)(5,7,9,10)(11,12)(13,18)(14,24)(15,19)(16,30)(17,21)(20,31)(22,32)(23,26)(25,27)(28,40)(29,35)(33,37)(34,41)(36,38)(39,42);(3,4)(5,7)(6,8)(9,10)(11,13,17,24)(12,18,21,14)(15,27,29,22)(16,23,31,37)(19,25,35,32)(20,33,30,26)(28,39,41,36)(34,38,40,42);(11,14)(12,13)(15,25)(16,26)(17,18)(19,22)(20,23)(21,24)(27,35)(28,38)(29,32)(30,37)(31,33)(34,36)(39,40)(41,42);(11,12)(13,18)(14,24)(15,19)(16,20)(17,21)(22,32)(23,33)(25,27)(26,37)(28,34)(29,35)(30,31)(36,42)(38,39)(40,41);(11,15,17,29)(12,19,21,35)(13,22,24,27)(14,25,18,32)(16,28,31,41)(20,34,30,40)(23,36,37,39)(26,38,33,42);(3,5,6,9)(4,7,8,10)(11,16)(12,20)(13,23)(14,26)(15,28)(17,31)(18,33)(19,34)(21,30)(22,36)(24,37)(25,38)(27,39)(29,41)(32,42)(35,40)164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493214);