Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138844 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,4,8,5,7,9,15)(3,6,11,13,10,14,12,16)(17,18,22,27,24,30,40,47,23,29,20,37,39,42,21,28)(19,31,34,45,36,43,33,44,35,48,32,26,41,46,38,25);(1,3,9,11,5,10,4,12)(2,6,8,16,7,14,15,13)(17,19)(18,25,29,44)(20,38)(21,32)(22,33)(23,35)(24,41)(26,42,45,30)(27,43,37,46)(28,48,47,31)(34,40)(36,39);(17,19)(18,26)(20,38)(21,32)(22,33)(23,35)(24,41)(25,42)(27,31)(28,43)(29,45)(30,44)(34,40)(36,39)(37,48)(46,47)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=gh,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=eghi,f^c=fghi,h^c=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46040);