Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94486 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7324 |
| commutator | C23×C4 | C42 |
| Frattini, cosocle | #2310 | C22 |
| prime radical | C24 | #95 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2310 ◁ G | |
| LECS | G ▷ #2310 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
265 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 40 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:
(1,2,5,10,7,12,26,37)(3,13,15,40,17,32,23,51)(4,18,20,35,22,9,24,33)(6,11,25,36,27,38,44,61)(8,28,29,19,31,45,56,14)(16,47,46,63,49,41,52,62)(21,39,50,60,42,34,53,59)(30,57,55,43,58,54,64,48);(1,3,6,16)(2,8,11,30,12,31,38,58)(4,19,21,48)(5,23,25,52)(7,17,27,49)(9,32,34,47,18,13,39,41)(10,29,36,55,37,56,61,64)(14,42,43,22)(15,44,46,26)(20,28,50,54)(24,45,53,57)(33,51,59,63,35,40,60,62)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=i,e^2=f,b^a=bc,c^a=cf,e^a=egi,f^a=fi,c^b=cg,d^b=dfi,g^b=gh,h^b=hi,d^c=di,e^c=eh,g^e=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72284166, #72284174, #72284191, … |
| maximal normal subgroup of | #72284166, #72284174, #72284191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1684);