Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#50942 |
| 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);(4,7)(8,10)(11,12,16,20)(13,14,18,17)(15,19,23,26)(21,22,25,24)(27,28,33,35)(29,36,30,31)(32,40,41,34)(37,39,38,42);(13,18)(14,17)(15,23)(19,26)(27,29,33,30)(28,31,35,36)(32,37,41,38)(34,39,40,42);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,20)(15,21)(16,18)(19,24)(22,26)(23,25)(27,30,33,29)(28,31,35,36)(32,38,41,37)(34,39,40,42);(11,14,16,17)(12,18,20,13)(15,22,23,24)(19,25,26,21)(27,31)(28,29)(30,35)(32,39)(33,36)(34,37)(38,40)(41,42);(3,5,6,9)(4,8,7,10)(11,15,16,23)(12,19,20,26)(13,21,18,25)(14,22,17,24)(27,32)(28,34)(29,37)(30,38)(31,39)(33,41)(35,40)(36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613835);