Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #329 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C22×#97 | C22 |
| socle | C22 | #1295 |
| prime radical | C24 | #101 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 2 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,15,20,14,19)(11,21,23,31,25,32,22,16)(13,27,26,18,28,33,29,34)(24,38,37,40,39,30,36,35);(1,3)(2,5,6,7)(4,8)(9,11,13,24,14,22,29,36,15,25,28,39,12,23,26,37)(10,16,18,30,17,21,33,35,20,31,34,38,19,32,27,40)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,h^a=hi,c^b=cg,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 575);