Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12747 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1786 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4)(5,16)(6,21,24,39)(7,25)(8,14)(9,31)(10,15)(11,18)(12,23,38,30)(13,28)(17,45,48,65)(19,43)(20,44)(22,37,55,63)(26,34)(27,41)(29,57)(32,40)(33,60,61,53)(35,58)(36,50)(42,47,64,49)(46,51,66,62)(52,59,56,54);(3,5)(4,11)(6,17)(7,26)(8,19)(9,32)(10,20)(12,33)(13,40)(14,34)(15,36)(16,25)(18,28)(21,51)(22,53)(23,47)(24,48)(27,50)(29,58)(30,49)(31,35)(37,45)(38,61)(39,62)(41,43)(42,56)(44,57)(46,54)(52,64)(55,60)(59,66)(63,65);(3,6)(4,12)(5,17)(7,22)(8,30)(9,23)(10,24)(11,33)(13,37)(14,39)(15,38)(16,42)(18,46)(19,49)(20,48)(21,31)(25,56)(26,53)(27,59)(28,54)(29,55)(32,47)(34,62)(35,51)(36,61)(40,45)(41,63)(43,65)(44,64)(50,66)(52,57)(58,60)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400742);