Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#97 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
322 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 198 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:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,20,26,22,25,32,30,24,27,31,29,23,28,21,19);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,28,23,29,31,27,24,30,32,25,22,26,20,18);(3,9)(6,12)(10,15)(13,16)(18,25,27,28)(19,29,30,26)(20,31)(21,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,h^2=i,c^a=ch,f^a=fi,c^b=cg,f^b=fi,d^c=di,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72227016, #72227019, #72227020, … |
| maximal normal subgroup of | #72227016, #72227019, #72227020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33736);