Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12753 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3532 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,11)(6,14,15,9)(7,10)(8,12,16,18)(13,17);(3,5)(4,8)(6,12)(7,13)(9,11)(10,16)(14,17)(15,18);(9,14)(11,17)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213329, C2×#511790, D3×#12753, … |
| maximal normal subgroup of | #72213329, C2×#511790, #72218088, … |
| maximal quotient of | #72210566, #72210595, D3×#12753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400755);