Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| 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 | |
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) = 256 |
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)(17,18,23,28,24,27,41,46,21,29,36,42,22,37,39,47)(19,20,31,35,33,34,26,44,32,25,40,43,30,38,45,48);(1,3,6,2,7,11,5,9)(4,10,13,8,15,12,14,16)(17,19,23,40,24,33,41,45,21,32,36,31,22,30,39,26)(18,25,42,43,27,38,47,48,29,20,28,35,37,34,46,44);(1,4,7,15)(2,8,9,16)(3,10,11,12)(5,14,6,13)(17,20,21,25)(18,26,29,45)(19,28,32,42)(22,38,24,34)(23,35,36,43)(27,40,37,31)(30,47,33,46)(39,48,41,44)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=di,e^b=egi,f^b=fi,d^c=dgi,e^c=eg,f^c=fgi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45055);