Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #117993 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1924 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6)(3,10)(4,7)(8,15)(9,13)(12,16)(17,18,22,28,21,26,30,25,23,27,39,45,37,46,20,35)(19,29,31,44,33,24,36,42,34,43,38,47,32,41,48,40);(1,2,7,13,5,9,4,6)(3,8,14,15,10,16,11,12)(17,19,30,48,37,32,22,38,23,34,20,36,21,33,39,31)(18,24,25,44,46,29,28,40,27,41,35,47,26,43,45,42);(1,3,4,11,5,10,7,14)(2,8,6,12,9,16,13,15)(17,20,21,22,23,30,37,39)(18,25,26,45,27,35,46,28)(19,31,32,48,34,38,33,36)(24,40,29,47,41,42,43,44)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ch,d^b=di,f^b=fi,e^c=eg,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47361);