Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7410 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1344 |
| 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 | |
349 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 349 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:
(1,2,5,10,7,13,25,12)(3,14,16,38,19,40,18,50)(4,20,21,9,23,32,53,34)(6,26,24,11,27,36,44,37)(8,28,29,15,31,41,35,43)(17,46,45,39,49,63,48,58)(22,54,52,51,55,33,61,60)(30,42,56,64,59,47,62,57);(1,3,6,17)(2,8,11,30)(4,15,22,42)(5,18,24,48)(7,19,27,49)(9,14,33,39)(10,35,36,62)(12,29,26,56)(13,31,37,59)(16,44,45,25)(20,38,51,63)(21,28,52,57)(23,43,55,47)(32,50,60,46)(34,40,54,58)(41,61,64,53)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=i,f^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,c^b=cg,d^b=di,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, 1485);