Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #357 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | C4×#357 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,6,14,10,17,9,37)(3,18,21,51,25,45,24,59)(4,12,27,16,8,36,29,48)(5,30,32,61,33,15,28,44)(7,13,34,46,26,47,35,62)(11,38,40,31,43,55,63,20)(19,49,52,41,23,58,53,60)(22,39,56,42,54,50,57,64);(1,3,7,22,8,23,32,55,6,21,34,56,29,53,33,20,10,25,26,54,4,19,28,38,9,24,35,57,27,52,5,31)(2,11,15,41,16,42,46,59,37,63,61,49,12,39,13,45,17,43,30,60,48,64,62,51,14,40,44,58,36,50,47,18);(1,4,6,27,10,8,9,29)(2,12,14,16,17,36,37,48)(3,19,21,52,25,23,24,53)(5,26,32,35,33,7,28,34)(11,39,40,42,43,50,63,64)(13,44,46,30,47,61,62,15)(18,49,51,41,45,58,59,60)(20,22,38,56,31,54,55,57)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gh,e^2=i,f^2=g,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72223679, C2×#151968, #72225311, … |
| maximal normal subgroup of | #72223679, C2×#151968, #72225311, … |
| maximal quotient of | C8×#1419, C8×#1418, C2×#151968 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59164);