Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
292 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,6)(4,12,15,10)(5,11)(7,9)(8,14,16,13)(17,18,21,27,23,28,38,36,22,26,29,43,39,19,24,31)(20,32,34,44,35,30,40,45,33,25,37,41,48,42,47,46);(1,3,6,2,7,11,5,9)(4,10,13,8,15,12,14,16)(17,19,29,31,23,18,24,27,22,28,21,36,39,26,38,43)(20,30,34,46,35,25,40,44,33,42,37,45,48,32,47,41);(1,4,7,15)(2,8,9,16)(3,10,11,12)(5,14,6,13)(17,20,22,33)(18,25,26,32)(19,30,28,42)(21,37,29,34)(23,35,39,48)(24,40,38,47)(27,44,43,41)(31,46,36,45)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=di,e^b=eg,f^b=fi,d^c=dg,e^c=egi,f^c=fg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45053);