Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150413 |
| 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 | |
409 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 409 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:
(1,2,6,11,7,13,30,42)(3,14,17,46,18,47,51,63)(4,20,21,53,23,54,55,64)(5,24,27,41,22,10,38,39)(8,31,29,59,26,12,44,43)(9,33,35,25,37,56,61,49)(15,36,48,62,16,45,50,40)(19,52,28,57,32,34,58,60);(1,3)(2,9)(4,16)(5,25)(6,28)(7,18)(8,32)(10,34)(11,40)(12,45)(13,37)(14,20)(15,23)(17,29)(19,26)(21,33)(22,49)(24,52)(27,50)(30,58)(31,36)(35,59)(38,48)(39,63)(41,46)(42,62)(43,61)(44,51)(47,54)(53,60)(55,56)(57,64);(1,4,8,22,7,23,26,5)(2,10,12,20,13,24,31,54)(3,15,19,49,18,16,32,25)(6,21,29,38,30,55,44,27)(9,34,36,47,37,52,45,14)(11,39,43,53,42,41,59,64)(17,48,28,33,51,50,58,56)(35,60,62,63,61,57,40,46)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=cfi,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=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57999);