Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7114 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | D4⋊C8 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,15,20,17,21,26,19,18,22,14,25,16,13,23,24);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,19)(15,24)(16,22)(18,21)(20,26)(23,25)116 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11710948, C22×#7114, #72213037, … |
| maximal normal subgroup of | C22×#7114, #72213037, #72213160, … |
| maximal quotient of | C2×#93386, #72211373, C2×#92837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31269);