Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1954 |
| prime radical | C23 | #436 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,7,12,21)(4,14)(5,18)(6,9)(8,16)(10,15)(11,23,25,32)(13,27,24,20)(17,28)(19,30,31,26)(22,29)(33,34,38,43,39,42,45,58,37,41,52,57,36,50,51,54)(35,44,48,60,49,55,61,64,47,59,62,63,46,40,53,56);(1,3,6,12)(2,7)(4,11,17,25)(5,13,15,24)(8,19)(9,21)(10,20)(14,23)(16,26,29,30)(18,27)(22,31)(28,32)(33,35,45,61,36,46,38,48,37,47,51,53,39,49,52,62)(34,40,54,63,50,59,57,64,41,55,58,60,42,44,43,56);(1,4,15,29,6,17,5,16)(2,8,18,28,9,22,10,14)(3,11,24,30,12,25,13,26)(7,19,27,32,21,31,20,23)(34,41)(40,55)(42,50)(43,57)(44,59)(54,58)(56,64)(60,63)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dgi,f^b=fgi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48013);