Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#36701 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| socle, prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ 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);(3,4)(5,8)(6,7)(9,10)(11,12)(13,17)(14,18)(15,16)(19,25)(20,23)(21,26)(22,24)(27,28,31,34)(29,32,36,40)(30,33,38,42)(35,39,37,41);(12,16)(17,23)(18,24)(25,26)(27,29,31,36)(28,32,34,40)(30,37,38,35)(33,41,42,39);(4,7)(8,10)(11,13,15,20)(12,17,16,23)(14,21,22,19)(18,26,24,25)(27,30)(28,33)(29,35)(31,38)(32,39)(34,42)(36,37)(40,41);(11,14,15,22)(12,18,16,24)(13,19,20,21)(17,25,23,26)(28,34)(30,38)(32,40)(35,37);(3,5,6,9)(4,8,7,10)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540605);