Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1677 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
286 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,12)(7,17)(8,13)(9,24)(10,23)(15,35)(16,25)(18,19)(20,22)(21,29)(26,31)(27,28)(30,36)(32,33);(5,6,13,28,10,15,9,25,11,16,24,35,23,27,8,12)(7,14,29,31,19,32,18,30,21,34,17,36,20,33,22,26);(5,7)(6,14)(8,22)(9,18)(10,19)(11,21)(12,26)(13,29)(15,32)(16,34)(17,24)(20,23)(25,30)(27,33)(28,31)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=dh,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^b=ei,d^c=di,e^c=eh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42186);