Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13584 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13591 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4⋊Q8 | C24 |
| prime radical | C2×#3151 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1274 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,11,8,12,15,18)(5,10,13,17,7,9,14,16);(4,9)(5,7)(6,14)(10,12)(13,15)(16,17);(3,5,8,7)(4,10,12,9)(6,13,15,14)(11,17,18,16)56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279151, C2×#153467, #72279237, … |
| maximal normal subgroup of | #72279151, C2×#153467, #72279237, … |
| maximal quotient of | #72279932, #72293202, #72293210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419802);