Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| socle | C2 | #11873 |
| prime radical | #24287 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 generators on 64 points:
(1,2,6,12,8,13,29,45)(3,15,18,48,20,49,53,62)(4,21,22,55,23,42,56,63)(5,11,27,41,24,10,38,39)(7,31,30,58,26,14,46,44)(9,33,35,51,36,59,61,25)(16,47,50,64,17,37,52,43)(19,34,54,60,32,40,28,57);(1,3)(2,9)(4,17)(5,25)(6,28)(7,32)(8,20)(10,34)(11,40)(12,43)(13,36)(14,47)(15,42)(16,23)(18,46)(19,26)(21,49)(22,33)(24,51)(27,50)(29,54)(30,53)(31,37)(35,44)(38,52)(39,62)(41,48)(45,64)(55,57)(56,59)(58,61)(60,63);(1,4,7,24,8,23,26,5)(2,10,14,21,13,11,31,42)(3,16,19,51,20,17,32,25)(6,22,30,38,29,56,46,27)(9,34,37,15,36,40,47,49)(12,39,44,55,45,41,58,63)(18,50,54,59,53,52,28,33)(35,60,43,48,61,57,64,62)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=dh,f^c=fi,e^d=eh,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57998);