Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #327 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C2 | #7029 |
| prime radical | C24 | #108 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,5,12,8,13,29,42,9,14,30,43,7,33,6,31)(3,15,17,45,21,47,52,64,20,38,51,46,19,55,18,53)(4,22,24,56,27,41,25,40,28,58,60,57,26,11,32,39)(10,34,35,50,36,48,62,23,37,59,63,49,44,54,61,16);(1,3,6,18,7,19,30,51,9,20,29,52,8,21,5,17)(2,10,12,35,13,36,42,62,14,37,43,63,33,44,31,61)(4,23,25,54,26,50,24,59,28,16,32,48,27,49,60,34)(11,38,40,45,22,55,57,64,41,15,39,46,58,47,56,53)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=h,e^2=f,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,c^b=cg,g^b=gi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 633);