Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | #33713 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
324 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6,9,15,31,47,28,44,11,17,29,45,10,16,32,48,12,18,33,49,26,42,57,64,35,51,8,14,27,43,62,68)(7,19,21,38,58,46,61,40,23,53,20,50,22,41,30,52,24,54,55,66,34,63,25,13,36,37,56,65,59,67,60,39);(5,7,11,36,12,24,35,23)(6,13,17,52,18,40,51,39)(8,25,9,30,29,61,33,60)(10,34,26,22,27,58,31,59)(14,41,15,46,45,67,49,63)(16,50,42,38,43,65,47,66)(19,48,37,64,54,68,53,44)(20,32,21,57,56,62,55,28);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=gh,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60184);