Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| 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 | C4×D16 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6,11,18,13,21,42,20)(7,22,26,61,27,48,33,63)(8,28,32,62,34,55,65,67)(9,35,38,46,10,40,41,68)(12,43,36,19,44,58,37,60)(14,45,49,25,52,66,51,24)(15,53,56,30,57,64,50,31)(16,23,47,29,17,59,54,39);(1,2,3,4)(5,7,23,64,44,34,10,25,42,33,39,30,36,32,38,45,13,27,59,53,12,8,9,24,11,26,29,31,37,65,41,66)(6,14,46,55,60,56,16,22,18,49,40,67,43,57,47,61,21,52,68,28,19,50,17,48,20,51,35,62,58,15,54,63);(5,8,29,45,12,33,38,64,42,65,23,24,37,27,9,30,13,34,39,66,44,26,41,53,11,32,59,25,36,7,10,31)(6,15,40,22,19,52,54,62,18,56,68,61,58,51,16,55,21,57,35,48,60,14,47,67,20,50,46,63,43,49,17,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286831, #72286872, #72320842, … |
| maximal normal subgroup of | #72286831, #72286872, #72320842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60172);