Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46191 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(4,7)(8,10)(11,12,16,19)(13,14,20,17)(15,18,23,25)(21,22,26,24)(27,28,33,31)(29,36,30,34)(32,39,40,35)(37,41,38,42);(13,20)(14,17)(15,23)(18,25)(27,29,33,30)(28,34,31,36)(32,37,40,38)(35,41,39,42);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,19)(15,21)(16,20)(18,24)(22,25)(23,26)(27,30,33,29)(28,34,31,36)(32,38,40,37)(35,41,39,42);(11,14,16,17)(12,13,19,20)(15,22,23,24)(18,21,25,26)(27,31,33,28)(29,34,30,36)(32,39,40,35)(37,41,38,42);(3,5,6,9)(4,8,7,10)(11,15,16,23)(12,18,19,25)(13,21,20,26)(14,22,17,24)(27,32)(28,35)(29,37)(30,38)(31,39)(33,40)(34,41)(36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609957);