Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112946 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1955 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 279 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,10,7,9,5,3)(4,12,14,16,15,11,13,8)(17,18,22,28,23,27,39,43,24,19,25,31,21,30,38,46)(20,32,36,47,37,29,44,45,34,33,40,48,35,26,41,42);(1,3,6,9,7,10,5,2)(4,12,14,8,15,11,13,16)(17,19,22,31,23,30,39,46,24,18,25,28,21,27,38,43)(20,33,40,47,37,26,41,45,34,32,36,48,35,29,44,42);(1,4,6,14,7,15,5,13)(2,8,3,11,9,16,10,12)(17,20,21,35,24,34,23,37)(18,26,27,32,19,29,30,33)(22,36,38,41,25,40,39,44)(28,42,43,47,31,45,46,48)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=eg,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ch,d^b=dg,e^b=ei,f^b=fg,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48020);