Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486811 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
737 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 737 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 4 generators on 64 points:
(1,2,6,11)(3,13,17,35)(4,9,20,33)(5,22,26,10)(7,12,27,37)(8,23,30,16)(14,38,40,60)(15,39,44,62)(18,31,21,34)(19,48,50,32)(24,54,47,61)(25,51,55,36)(28,49,56,41)(29,53,58,43)(42,63,46,64)(45,57,52,59);(1,3,6,17)(2,8,11,30)(4,14,20,40)(5,23,26,16)(7,15,27,44)(9,28,33,56)(10,35,22,13)(12,29,37,58)(18,46,21,42)(19,49,50,41)(24,52,47,45)(25,53,55,43)(31,59,34,57)(32,60,48,38)(36,62,51,39)(54,63,61,64);(1,4,7,21)(2,9,12,34)(3,14,15,42)(5,24,25,19)(6,20,27,18)(8,28,29,57)(10,32,36,61)(11,33,37,31)(13,38,39,63)(16,45,43,41)(17,40,44,46)(22,48,51,54)(23,52,53,49)(26,47,55,50)(30,56,58,59)(35,60,62,64);(3,15)(4,18)(8,29)(9,31)(13,39)(14,40)(16,43)(17,44)(19,47)(20,21)(23,53)(24,50)(28,56)(30,58)(32,54)(33,34)(35,62)(38,60)(41,49)(42,46)(45,52)(48,61)(57,59)(63,64)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394393);