Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13495 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,14,19,27,37,41,36,16,21,28,35,15,20,26,18)(13,17,23,30,24,31,40,42,25,33,39,29,22,32,38,34);(4,7)(8,10)(11,13)(12,17)(14,23)(15,22)(16,25)(18,34)(19,30)(20,32)(21,33)(24,27)(26,38)(28,39)(29,35)(31,37)(36,42)(40,41);(3,5,6,9)(4,8,7,10)(12,18)(13,22)(14,26)(15,27)(17,29)(19,20)(21,36)(23,39)(24,25)(28,41)(30,33)(31,42)(32,34)(35,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406575);