Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | C2×#121 | C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
580 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 112 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)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,21,27,23,28,37,44,22,26,36,42,20,25,34,41)(19,29,31,43,30,40,45,48,32,24,35,39,33,38,46,47);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,24)(20,33)(21,35)(22,32)(23,30)(25,40)(26,29)(27,43)(28,38)(31,36)(34,45)(37,46)(39,42)(41,47)(44,48);(1,3,5,12)(2,7,9,10)(4,11,13,16)(6,14,8,15)(17,20,22,23)(18,25,26,28)(19,30,32,33)(21,34,36,37)(24,38,29,40)(27,41,42,44)(31,45,35,46)(39,47,43,48)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ci,d^b=dgi,f^b=fi,f^c=fgi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36903);