Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150788 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C23 | #357 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,3,6,4,5,7,8)(9,10,13,19,36,51,61,49,16,22,12,29,15,21,37,53,14,20,34,52,31,47,33,57,39,54,32,18,38,48,69,72)(11,23,25,58,64,50,68,17,28,42,24,60,27,46,65,45,26,59,63,44,40,70,30,43,41,71,62,55,66,67,35,56);(2,5)(3,7)(9,11,14,26)(10,17,20,43)(12,30,32,68)(13,35,34,65)(15,40,38,64)(16,41,39,28)(18,46,29,67)(19,50,52,70)(21,55,48,60)(22,56,54,45)(23,57,59,49)(24,61,62,33)(25,37,63,69)(27,36,66,31)(42,53,71,72)(44,51,58,47);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=gh,e^2=fgh,g^2=hi,h^2=i,b^a=bd,c^a=cfi,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58180);