Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#12773 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C22 | #1790 |
| prime radical | C2×#3483 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2685 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 2685 conjugacy classes of subgroups — too many to draw. Only its 314 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)(5,9,6,11)(7,12)(8,13)(10,14,18,15)(16,17);(4,9)(5,6)(10,12)(11,13)(14,15)(17,18);(3,5)(4,10)(6,8)(7,15)(9,17)(11,12)(13,18)(14,16)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#511838, #72261011, #72261027, … |
| maximal normal subgroup of | C2×#511838, #72261011, #72261027, … |
| maximal quotient of | #72261884, C2×#149484 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400800);