Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33687 |
| 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 | |
1482 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)(11,12,15,20)(13,23,25,18)(14,28,29,64)(16,34,32,63)(17,37,33,22)(19,42,43,30)(21,46,45,36)(24,52,53,62)(26,58,56,67)(27,41,57,51)(31,65,38,47)(35,48,66,44)(39,54,49,71)(40,60,50,72)(55,69,61,73)(59,68,74,70);(4,7)(8,10)(11,13)(12,18)(14,24)(15,25)(16,26)(17,27)(19,39)(20,23)(21,40)(22,41)(28,62)(29,53)(30,54)(31,55)(32,56)(33,57)(34,67)(35,59)(36,60)(37,51)(38,61)(42,71)(43,49)(44,68)(45,50)(46,72)(47,69)(48,70)(52,64)(58,63)(65,73)(66,74);(11,14,15,29)(12,19,20,43)(13,24,25,53)(16,35,32,66)(17,38,33,31)(18,39,23,49)(21,47,45,65)(22,48,37,44)(26,59,56,74)(27,61,57,55)(28,63,64,34)(30,46,42,36)(40,69,50,73)(41,70,51,68)(52,67,62,58)(54,72,71,60);(3,5,6,9)(4,8,7,10)(13,25)(18,23)(24,53)(26,56)(27,57)(39,49)(40,50)(41,51)(52,62)(54,71)(55,61)(58,67)(59,74)(60,72)(68,70)(69,73)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530290);