Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #353 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | #2931 | C22 |
| socle | C2 | #7205 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,4,10,6,12,22,36,7,13,23,38,28,9,21,33)(3,14,15,35,17,8,19,29,18,31,45,57,50,32,44,59)(5,24,25,37,26,34,42,56,27,11,20,39,52,40,53,61)(16,46,47,62,48,54,55,58,49,41,43,63,64,30,51,60);(1,3,5,16,6,17,26,48,7,18,27,49,28,50,52,64)(2,8,11,30,12,31,40,46,13,32,24,54,9,14,34,41)(4,19,20,51,22,45,53,47,23,44,25,55,21,15,42,43)(10,35,37,62,36,29,56,58,38,57,39,63,33,59,61,60)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=g,e^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fh,h^a=hi,d^b=dfh,d^c=dhi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1356);