Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57827 |
| 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 | |
4620 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);(3,4,8,9)(5,7,6,10)(11,12,15,18)(13,19,20,16)(14,17,22,24)(21,25,26,23)(27,28)(29,31)(30,37)(32,34)(33,35)(36,39)(38,41)(40,42);(4,9)(5,6)(13,20)(16,19)(21,26)(23,25)(27,29,33,30)(28,31,35,37)(32,38,40,36)(34,41,42,39);(3,5,8,6)(4,10,9,7)(11,13)(12,16)(14,21)(15,20)(17,23)(18,19)(22,26)(24,25)(27,30,33,29)(28,31,35,37)(32,38,40,36)(34,39,42,41);(3,6,8,5)(4,7,9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,31)(28,30)(29,35)(32,39)(33,37)(34,38)(36,42)(40,41);(3,7)(4,5)(6,9)(8,10)(11,14,15,22)(12,17,18,24)(13,21,20,26)(16,23,19,25)(27,32,33,40)(28,34,35,42)(29,36,30,38)(31,39,37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008654);