Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | C23 | #405 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
267 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 122 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:
(1,2,4,7)(3,9,10,6)(5,13,12,8)(11,14,16,15)(17,18,21,27,22,26,35,42,23,28,36,43,20,24,34,38)(19,29,31,44,32,37,46,47,33,25,39,41,30,40,45,48);(1,3,5,11,4,10,12,16)(2,6,8,14,7,9,13,15)(18,24,28,26)(25,37,29,40)(27,38,43,42)(41,47,44,48);(17,19,21,31,22,32,35,46,23,33,36,39,20,30,34,45)(18,25,27,41,26,40,42,48,28,29,43,44,24,37,38,47)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=i,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fgi,d^b=dg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35708);