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 | #532 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 274 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) = 256 |
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)(17,18,22,28,24,27,42,44,23,29,41,32,21,40,31,37)(19,30,26,46,34,20,36,39,35,38,47,48,33,25,43,45);(1,3,6,9,7,10,5,2)(4,12,14,8,15,11,13,16)(17,19,31,43,21,33,41,47,23,35,42,36,24,34,22,26)(18,25,44,46,40,38,28,45,29,20,37,48,27,30,32,39);(1,4,6,14,7,15,5,13)(2,8,3,11,9,16,10,12)(17,20,24,38,23,25,21,30)(18,26,40,43,29,47,27,36)(19,32,33,44,35,28,34,37)(22,39,42,48,41,45,31,46)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=eg,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=dgi,e^b=ei,f^b=fgi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48021);