Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
320 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18,23,28,24,27,42,46,21,29,30,31,22,38,40,47)(19,20,33,37,35,36,26,44,34,25,41,43,32,39,45,48);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19,30,33,24,35,40,26,21,34,23,41,22,32,42,45)(18,25,28,37,27,39,46,44,29,20,31,43,38,36,47,48);(2,8)(4,5)(7,15)(11,13)(17,20)(18,26)(19,31)(21,25)(22,39)(23,37)(24,36)(27,41)(28,34)(29,45)(30,43)(32,46)(33,38)(35,47)(40,48)(42,44)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=di,e^b=egi,f^b=fi,d^c=dgi,e^c=ei,f^c=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45084);