Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1939 |
| prime radical | C23 | C8×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,10)(4,5)(8,15)(9,13)(11,12)(17,18,22,26,23,19,27,29,21,25,30,32,20,28,24,31);(1,2)(3,8)(4,13)(5,7)(6,9)(10,14)(11,16)(12,15)(17,19,22,29,23,25,27,32,21,28,30,31,20,18,24,26);(1,3,4,11,6,12,5,10)(2,8,13,16,9,15,7,14)(17,20,21,23)(18,19,25,28)(22,24,30,27)(26,29,32,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=eg,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ci,d^b=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47610);