Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1850 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,21,13,20,30,39,16,22,24,34,14,19,32,38)(11,23,26,28,18,35,36,29,27,37,40,31,25,17,33,12);(2,6)(3,4)(7,8)(9,11,24,40,13,18,32,33,16,27,15,26,14,25,30,36)(10,17,34,29,20,23,38,31,22,35,21,12,19,37,39,28);(2,7)(3,4)(6,8)(9,12)(10,18)(11,19)(13,28)(14,31)(15,23)(16,29)(17,32)(20,27)(21,36)(22,25)(24,37)(26,38)(30,35)(33,34)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=egi,d^c=dgi,e^c=eg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43098);