Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94530 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7339 |
| commutator | C2×C22⋊C4 | C2×C8 |
| Frattini, cosocle | #1822 | C22 |
| prime radical | #534 | C8 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1822 ◁ G | |
| LECS | G ▷ #1822 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 24 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:
(1,2,5,10,6,12,22,35,7,11,23,36,25,37,52,60)(3,13,15,40,16,41,44,62,17,30,45,54,47,56,63,64)(4,9,18,31,20,32,49,57,19,33,21,50,24,39,51,59)(8,26,27,43,29,53,55,42,28,48,34,46,38,61,58,14);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,30)(10,34)(11,28)(12,38)(13,33)(18,48)(19,42)(20,43)(21,26)(22,44)(23,45)(24,46)(25,47)(27,36)(29,37)(31,54)(32,41)(35,55)(39,56)(40,50)(49,61)(51,53)(52,63)(57,62)(58,60)(59,64)Polycyclic presentation:
2^9|a^2=d,c^2=h,d^2=f,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=deh,f^b=fh,g^b=gi,d^c=dg,e^c=ei,f^c=fi,e^d=eh,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1727);