Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9558 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,6,14,8,15,35,13,9,16,34,12,7,37,5,32)(3,17,21,50,23,39,33,10,24,40,20,41,22,56,19,42)(4,25,28,46,30,48,27,57,31,47,36,11,29,44,59,45)(18,38,53,49,26,43,51,62,55,61,52,63,54,64,58,60);(1,3,7,22,9,24,8,23)(2,10)(4,26,30,55,31,54,29,18)(5,33,34,21,35,19,6,20)(11,43,46,64)(12,40)(13,39)(14,17)(15,41)(16,42)(25,49,47,63)(27,53,36,51,59,52,28,58)(32,56)(37,50)(38,45,61,57)(44,60,48,62);(1,4,5,27,7,29,34,28,9,31,35,59,8,30,6,36)(2,11,12,44,15,45,32,25,16,46,14,48,37,57,13,47)(3,18,19,51,22,54,20,53,24,55,33,58,23,26,21,52)(10,38,17,49,41,43,39,62,42,61,40,63,50,64,56,60)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ch,d^b=di,f^b=fi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224954, #72224958, #72224964, … |
| maximal normal subgroup of | #72224954, #72224958, #72224964, … |
| maximal quotient of | C2×#140204 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47400);