Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10370 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #124 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1985 |
| prime radical | C23 | #407 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10,15,20,13,19,31,23,16,21,33,11,14,18,32,24)(12,25,28,26,30,36,40,22,29,37,39,38,27,17,34,35);(1,3,7,2,5,8,4,6)(9,11,15,18,13,24,31,10,16,20,33,19,14,23,32,21)(12,26,39,17,30,22,34,25,29,38,28,36,27,35,40,37);(2,6)(3,8)(9,12,13,30,16,29,14,27)(10,17,18,37,21,36,19,25)(11,22,23,26,20,35,24,38)(15,28,31,40,33,39,32,34)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=eg,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cgh,d^b=di,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49121);