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 |
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×#366 |
| prime radical | C23 | C2×Q32 |
| 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 | |
186 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 186 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)(5,6,9,14,11,17,30,16)(7,18,20,48,23,49,22,55)(8,24,27,47,29,57,62,56)(10,31,26,15,33,43,61,45)(12,34,35,51,37,50,46,19)(13,38,40,63,42,65,41,28)(21,53,52,66,54,67,32,39)(25,44,58,64,60,68,59,36);(5,7,9,20,11,23,30,22)(6,12,14,35,17,37,16,46)(8,25,27,58,29,60,62,59)(10,32,26,21,33,52,61,54)(13,39,40,53,42,66,41,67)(15,44,43,64,45,68,31,36)(18,47,48,57,49,56,55,24)(19,38,34,63,51,65,50,28);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=e,d^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,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, 60061);