Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7398 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1343 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C2 | #7250 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,5,10,7,13,21,12)(3,14,16,40,18,33,44,42)(4,19,22,9,24,34,52,36)(6,26,25,11,27,37,53,38)(8,28,29,20,32,51,31,15)(17,47,46,59,49,62,63,41)(23,39,54,50,43,35,56,61)(30,45,58,57,60,48,55,64);(1,3,6,17,7,18,27,49)(2,8,11,30,13,32,38,60)(4,20,23,48,24,15,43,45)(5,16,25,46,21,44,53,63)(9,33,35,59,36,14,39,41)(10,29,37,58,12,31,26,55)(19,40,50,47,34,42,61,62)(22,51,54,64,52,28,56,57)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,c^b=cg,g^b=gh,h^b=hi,e^c=eh,g^e=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1482);