Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1850 |
| prime radical | #121 | C2×C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,7,12)(3,13,4,15)(5,11,6,10)(8,16,9,14)(17,18,22,26,21,28,38,31,23,27,39,19,29,25,43,32)(20,33,35,46,37,42,48,30,36,44,47,45,34,24,40,41);(1,3,7,4)(2,8,12,9)(5,16,6,14)(10,15,11,13)(17,19,22,25,21,32,38,18,23,26,39,28,29,31,43,27)(20,30,35,44,37,45,48,24,36,41,47,33,34,46,40,42);(1,4,7,3)(2,9,12,8)(5,16,6,14)(10,15,11,13)(17,20,23,36)(18,24,27,42)(19,30,26,41)(21,37,29,34)(22,35,39,47)(25,44,28,33)(31,46,32,45)(38,48,43,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=ei,d^c=di,e^c=eg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43070);