Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C2×D8 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | C2×D4⋊C16 |
| prime radical | C2×#135 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,5,11,7,13,30,42,6,14,29,43,4,23,26,54,8,15,31,44,27,10,37,39,25,12,45,41,28,40,56,62)(3,16,19,48,21,33,52,58,22,38,53,61,24,46,55,63,17,47,49,64,18,9,32,34,20,36,51,60,50,35,57,59);(1,3)(2,9)(4,24)(5,19)(6,22)(7,21)(8,17)(10,38)(11,34)(12,46)(13,36)(14,35)(15,33)(16,23)(18,27)(20,25)(26,55)(28,50)(29,53)(30,52)(31,49)(32,37)(39,61)(40,47)(41,63)(42,60)(43,59)(44,58)(45,51)(48,54)(56,57)(62,64);(3,17)(9,33)(16,47)(18,21)(19,49)(20,22)(24,50)(32,52)(34,58)(35,46)(36,38)(48,64)(51,53)(55,57)(59,63)(60,61)Polycyclic presentation:
2^9|a^2=e,d^2=fi,e^2=g,f^2=i,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,c^b=ci,d^b=df,f^b=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225353, #72225379, #72225381, … |
| maximal normal subgroup of | #72225353, #72225379, #72225381, … |
| maximal quotient of | #72226992 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57247);