Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7394 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1343 |
| commutator | C2×C22⋊C4 | C2×C8 |
| Frattini, cosocle | #1729 | C22 |
| socle | C2 | #7123 |
| prime radical | C24 | C22⋊C8 |
| 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 | |
231 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 231 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:
(1,2,5,10,7,12,22,34,6,11,21,33,23,35,49,59)(3,13,15,37,17,39,43,62,16,38,44,55,24,51,50,64)(4,9,18,29,20,31,46,56,19,30,45,32,47,57,63,58)(8,25,26,48,27,52,54,14,28,40,53,41,36,61,60,42);(1,3)(2,8)(4,14,19,42)(5,15,21,44)(6,16)(7,24)(9,13,30,38)(10,26,33,53)(11,28)(12,36)(17,23)(18,25)(20,48,47,41)(22,50,49,43)(27,35)(29,55)(31,51,57,39)(32,37)(34,60,59,54)(40,45)(46,61)(52,63)(56,62)(58,64)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fhi,d^2=g,g^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defhi,e^b=ei,f^b=fi,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, 1162);