Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145540 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10536 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,14)(4,20)(5,25)(6,12)(7,31)(8,13)(9,16)(10,23)(11,28)(15,43)(17,37)(18,49)(19,42)(21,53)(22,55)(24,52)(26,39)(27,38)(29,40)(30,36)(32,59)(33,46)(34,47)(35,58)(41,45)(44,61)(48,64)(50,54)(51,57)(56,62)(60,63);(1,3,8,19)(2,9,13,34)(4,21,24,15)(5,26,29,46)(6,18,7,17)(10,32,36,62)(11,37,38,49)(12,39,31,33)(14,40,42,25)(16,27,47,28)(20,50,52,61)(22,56,58,59)(23,45,30,48)(35,43,55,53)(41,63,64,51)(44,57,54,60);(1,4,5,22,6,30,27,60,8,24,29,58,7,23,28,57)(2,10,11,35,31,52,40,63,13,36,38,55,12,20,25,51)(3,15,16,44,17,48,46,56,19,21,47,54,18,45,26,59)(9,32,14,41,39,50,49,53,34,62,42,64,33,61,37,43)Polycyclic presentation:
2^9|b^2=i,c^2=d,d^2=gi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fhi,g^b=gi,e^c=egh,f^c=fghi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52737);