Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41582 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4600 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);(5,9)(8,10)(11,12,16,17)(13,20,18,14)(15,19,23,24)(21,26,25,22)(27,28)(29,36)(30,31)(32,33)(34,39)(35,41)(37,38)(40,42);(3,4,7,6)(5,8,9,10)(12,17)(13,18)(19,24)(21,25)(28,33)(31,35)(34,36)(38,42);(3,5,7,9)(4,8,6,10)(11,13,16,18)(12,14,17,20)(15,21,23,25)(19,22,24,26)(27,29,32,39)(28,34,33,36)(30,40,41,37)(31,38,35,42);(3,6,7,4)(5,10,9,8)(11,14)(12,18)(13,17)(15,22)(16,20)(19,25)(21,24)(23,26)(27,30)(28,35)(29,37)(31,33)(32,41)(34,38)(36,42)(39,40);(11,15,16,23)(12,19,17,24)(13,21,18,25)(14,22,20,26)(27,31,32,35)(28,30,33,41)(29,38,39,42)(34,40,36,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547345);