Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153108 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2246 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13325 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
370 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,16)(4,22)(5,29)(6,35)(7,38)(8,13)(9,17)(10,18)(11,27)(12,50)(14,25)(15,37)(19,61)(20,40)(21,56)(23,45)(24,64)(26,52)(28,63)(30,55)(31,42)(32,48)(33,53)(34,54)(36,46)(39,43)(41,59)(44,58)(47,62)(49,57)(51,60);(1,3)(2,9)(4,23)(5,30)(6,36)(7,39)(8,21)(10,38)(11,46)(12,47)(13,41)(14,45)(15,44)(16,22)(17,27)(18,33)(19,25)(20,37)(24,28)(26,60)(29,43)(31,34)(32,62)(35,42)(40,54)(48,64)(49,51)(50,58)(52,55)(53,61)(56,63)(57,59);(1,3)(2,10)(4,24)(5,31)(6,36)(7,39)(8,21)(9,15)(11,47)(12,51)(13,44)(14,55)(16,29)(17,57)(18,50)(19,25)(20,37)(22,40)(23,28)(26,60)(27,59)(30,34)(32,62)(33,58)(35,45)(38,41)(42,48)(43,63)(46,53)(49,61)(52,64)(54,56)Polycyclic presentation:
2^9|d^2=gh,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dgi,e^c=egi,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224986, #72224995, #72225007, … |
| maximal normal subgroup of | #72224986, #72224995, #72225007, … |
| maximal quotient of | C2×#153108 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60304);