Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152543 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13012 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2263 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,6,14,39,24,35,21,8,15,5,13,34,25,55,59,9,16,36,32,10,41,44,20,18,23,37,12,7,3,17,22)(4,26,28,47,40,56,33,11,30,46,63,19,43,51,38,48,31,45,54,57,61,52,62,49,42,60,27,58,29,50,64,53);(1,3,18,24,9,25,8,41)(2,10,15,34,16,39,23,7)(4,19,30,47,31,58,42,57)(5,32,6,12,37,14,36,13)(11,43,53,61,49,29,48,40)(17,22,44,21,55,59,35,20)(26,54,60,27,45,28,46,63)(33,56,38,50,62,52,64,51);(1,4)(2,11)(3,19)(5,33)(6,38)(7,40)(8,42)(9,31)(10,43)(12,50)(13,51)(14,52)(15,53)(16,49)(17,54)(18,30)(20,26)(21,45)(22,60)(23,48)(24,47)(25,58)(27,44)(28,55)(29,39)(32,56)(34,61)(35,63)(36,64)(37,62)(41,57)(46,59)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=ghi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=egi,f^b=fg,g^b=gh,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225562, #72225568, #72225572, … |
| maximal normal subgroup of | #72225562, #72225568, #72225572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59740);