Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7394 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1343 |
| commutator | C2×C22⋊C4 | C2×C8 |
| Frattini, cosocle | #1729 | C22 |
| socle | C2 | #7103 |
| prime radical | #3449 | C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ #1729 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,5,10,6,12,23,37,7,11,22,36,25,38,52,60)(3,13,15,40,16,41,45,62,17,30,21,50,24,53,51,64)(4,9,18,31,20,33,46,56,19,32,47,35,48,57,63,59)(8,26,27,43,29,54,55,42,28,44,34,49,39,61,58,14);(1,3)(2,8)(4,14)(5,21)(6,24)(7,17)(9,30)(10,34)(11,28)(12,39)(13,32)(15,22)(16,25)(18,26)(19,42)(20,49)(23,45)(27,36)(29,38)(31,50)(33,41)(35,40)(37,55)(43,48)(44,47)(46,61)(51,52)(53,57)(54,63)(56,62)(58,60)(59,64)Polycyclic presentation:
2^9|a^2=d,c^2=f,d^2=g,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=def,g^b=gi,h^b=hi,d^c=dh,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1098);