Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,21,28,24,30,23,26)(19,31,34,42,32,46,36,48)(20,37,39,45,40,29,22,27)(25,35,41,38,43,47,44,33);(1,3)(2,6,8,15)(4,11)(5,10)(7,9,12,16)(13,14)(17,19,22,35,23,36,40,33,24,32,39,47,21,34,20,38)(18,25,29,42,28,41,27,46,30,43,37,48,26,44,45,31);(3,10)(6,15)(9,16)(11,14)(18,26,30,28)(19,32)(25,41,43,44)(27,29,45,37)(31,42,46,48)(33,38)(34,36)(35,47)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=cgh,d^b=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35815);