Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1351 |
| commutator | C2×C22⋊C4 | C2×C8 |
| Frattini, cosocle | #1729 | C22 |
| socle | C2 | #7122 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ #1729 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,5,10,6,12,23,35,7,11,24,34,25,36,50,58)(3,13,15,37,17,39,43,59,16,38,42,52,44,60,62,64)(4,18,20,45,22,46,48,63,21,9,30,31,49,32,55,56)(8,26,27,47,28,51,53,14,29,40,33,41,54,61,57,19);(1,3)(2,8)(4,19,21,14)(5,15)(6,17)(7,16)(9,13,18,38)(10,33)(11,29)(12,28)(20,40,30,26)(22,47,49,41)(23,43)(24,42)(25,44)(27,34)(31,37,45,52)(32,39,46,60)(35,57)(36,54)(48,61,55,51)(50,62)(53,58)(56,59,63,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defh,f^b=fi,h^b=hi,d^c=dh,e^c=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1140);