Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7493 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1390 |
| commutator | C2×D4 | C4×C8 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C2 | #7187 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,29,43,8,13,30,45,6,31,28,44)(3,14,16,46,18,47,26,57,19,38,52,60,17,53,51,63)(4,20,22,54,24,10,37,39,25,40,27,58,23,41,56,64)(9,32,33,21,34,55,61,49,36,59,62,15,35,48,42,50);(1,3,6,17,8,19,7,18)(2,9,12,34,13,36,31,35)(4,21,23,50,25,15,24,49)(5,26,28,16,30,51,29,52)(10,38,40,53,41,14,20,47)(11,42,43,33,45,61,44,62)(22,32,56,48,27,59,37,55)(39,63,58,46,64,57,54,60)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1295);