Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141451 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1970 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | C4×D8 |
| 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 | |
258 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 84 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)(17,18,23,29,24,31,40,30)(19,26,34,46,35,43,48,47)(20,36,32,25,38,44,33,45)(21,39,41,27,22,42,37,28);(1,3,9,11,5,10,4,12)(2,6,8,16,7,14,15,13)(17,19,22,33,40,48,41,38,24,35,21,32,23,34,37,20)(18,25,27,26,29,44,42,46,31,45,28,43,30,36,39,47);(17,20,37,34,23,32,21,35,24,38,41,48,40,33,22,19)(18,26,42,45,30,47,27,44,31,43,39,25,29,46,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=egh,d^2=ghi,e^2=ghi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,d^b=dh,f^b=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48735);