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 | #7131 |
| 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 | |
267 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 267 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:
(1,2,5,10,7,12,24,36,6,11,23,37,25,38,52,60)(3,13,15,40,17,41,45,62,16,42,22,50,26,53,51,64)(4,9,19,31,21,32,48,56,20,33,47,35,49,57,63,59)(8,27,28,43,30,54,55,18,29,44,34,46,39,61,58,14);(1,3,6,16)(2,8)(4,18,20,14)(5,22,23,15)(7,26,25,17)(9,13)(10,34)(11,29)(12,39)(19,44,47,27)(21,43,49,46)(24,45,52,51)(28,37)(30,38)(31,40)(32,53)(33,42)(35,50)(36,55)(41,57)(48,54,63,61)(56,64)(58,60)(59,62)Polycyclic presentation:
2^9|a^2=d,b^2=f,c^2=fi,d^2=g,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=defi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1173);