Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33741 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 74 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 74 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,17,22)(13,18,24,37)(14,27,31,56)(15,32,34,60)(16,35,33,21)(19,40,42,29)(20,43,44,28)(23,45,47,67)(25,52,50,73)(26,53,51,39)(30,59,58,55)(36,61,62,48)(38,65,64,46)(41,57,54,66)(49,72,71,69)(63,70,68,74);(4,7)(8,10)(11,13,17,24)(12,18,22,37)(14,23,31,47)(15,25,34,50)(16,26,33,51)(19,36,42,62)(20,38,44,64)(21,39,35,53)(27,45,56,67)(28,46,43,65)(29,48,40,61)(30,49,58,71)(32,52,60,73)(41,63,54,68)(55,69,59,72)(57,70,66,74);(11,14,28,57,16,30,34,40,17,31,43,66,33,58,15,29)(12,19,32,55,21,41,44,56,22,42,60,59,35,54,20,27)(13,23,46,70,26,49,50,61,24,47,65,74,51,71,25,48)(18,36,52,69,39,63,64,67,37,62,73,72,53,68,38,45);(3,5,6,9)(4,8,7,10)(13,24)(18,37)(23,47)(25,50)(26,51)(36,62)(38,64)(39,53)(45,67)(46,65)(48,61)(49,71)(52,73)(63,68)(69,72)(70,74)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530430);