Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93529 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1284 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | C22×#131 | C22 |
| socle | C2 | #7035 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#131 ▷ C22×C4 ▷ C2 ▷ C1 | |
463 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 463 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 generators on 64 points:
(1,2,5,10)(3,14,16,43)(4,19,22,50)(6,13,27,40)(7,11,26,41)(8,28,29,47)(9,33,35,24)(12,42,39,21)(15,30,46,38)(17,34,48,61)(18,45,25,56)(20,31,51,59)(23,36,55,64)(32,54,60,52)(37,58,63,53)(44,57,62,49);(1,3)(2,8,11,30)(4,20,23,52)(5,25,26,16)(6,17)(7,18)(9,34)(10,38)(12,31,13,32)(14,19)(15,24,47,53)(21,49)(22,54)(27,57,42,48)(28,58)(29,41)(33,46)(35,62,63,61)(36,45)(37,44)(39,60)(40,59)(43,50,56,64)(51,55)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 725);