Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152371 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12866 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
253 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,14,16,8)(4,19)(5,25)(6,30)(7,10)(9,15,35,48)(11,22)(12,28)(13,27)(17,51,50,63)(18,34,31,46)(20,45,54,32)(21,39)(23,61)(24,44)(26,40,47,33)(29,41)(36,56,64,62)(37,55,43,49)(38,60)(42,57)(52,59)(53,58);(1,3)(2,8)(4,15)(5,17)(6,18)(7,16)(9,19)(10,14)(11,32)(12,33)(13,34)(20,22)(21,47)(23,49)(24,48)(25,63)(26,28)(27,31)(29,50)(30,46)(35,44)(36,52)(37,53)(38,45)(39,40)(41,51)(42,64)(43,61)(54,60)(55,58)(56,59)(57,62);(1,3)(2,9,10,35)(4,20)(5,26)(6,31)(7,16)(8,19,14,44)(11,40,38,33)(12,32,39,45)(13,43,30,37)(15,22)(17,28)(18,27)(21,50)(23,62)(24,54)(25,36,41,64)(29,47)(34,61,46,53)(42,63,52,51)(48,60)(49,57)(55,59)(56,58)Polycyclic presentation:
2^9|a^2=f,c^2=d,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,f^b=fi,e^c=eg,f^c=fi,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72224991, #72225003, #72225004, … |
| maximal normal subgroup of | #72224991, #72225003, #72225004, … |
| maximal quotient of | C2×#152371 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59567);