Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1842 |
| prime radical | C23 | #423 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,21,26,22,25,30,32,23,27,29,24,20,31,19,28);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15);(2,7)(6,14)(8,12)(9,15)(17,19,20,29,23,30,22,21)(18,24,25,28,27,26,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=eg,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#135732, #72210332, #72210337, … |
| maximal normal subgroup of | C2×#135732, #72210332, #72210337, … |
| maximal quotient of | C2×#135732 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42928);