Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125098 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,5)(4,7)(6,8)(10,14,15,16)(17,18,21,26,23,28,36,25,22,27,37,44,35,43,20,34)(19,29,31,40,33,42,47,46,32,45,48,24,38,39,30,41);(2,5)(3,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19,22,32)(18,24,27,40)(20,31,36,48)(21,30,37,47)(23,38,35,33)(25,42,34,39)(26,45,44,29)(28,46,43,41);(9,11,13,12)(10,14,15,16)(17,20,35,37,22,36,23,21)(18,25,43,26,27,34,28,44)(19,30,38,48,32,47,33,31)(24,29,46,39,40,45,41,42)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fg,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ci,d^a=df,c^b=ch,d^b=dg,e^b=efg,g^b=gi,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32314);