Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1667 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9614 |
| prime radical | C23 | #473 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,7,14,9,16,8,15)(3,17,20,41,22,46,21,53)(4,23,26,37,28,55,27,54)(5,13,6,12,31,42,30,43)(10,32,34,52,35,29,44,19)(11,36,38,57,40,24,39,25)(18,47,49,33,51,59,50,60)(45,58,61,64,62,56,63,48);(1,3,7,20,9,22,8,21)(2,10,14,34,16,35,15,44)(4,18,26,49,28,51,27,50)(5,29,6,19,31,32,30,52)(11,33,38,59,40,60,39,47)(12,41,42,46,43,53,13,17)(23,45,37,61,55,62,54,63)(24,56,25,48,36,58,57,64);(1,4,5,24,8,27,30,57,9,28,31,36,7,26,6,25)(2,11,12,37,14,38,42,55,16,40,43,54,15,39,13,23)(3,18,29,56,21,50,52,64,22,51,32,58,20,49,19,48)(10,33,41,61,34,59,46,62,35,60,53,63,44,47,17,45)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=eh,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,g^b=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225389, #72225401, #72225403, … |
| maximal normal subgroup of | #72225389, #72225401, #72225403, … |
| maximal quotient of | #72225492 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47634);