Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C22×M32 | C22 |
| socle | C22 | #1314 |
| prime radical | C24 | M32 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C22×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(5,6,9,15,10,17,29,42,11,14,28,38,31,39,55,62,12,16,30,41,32,43,56,64,8,23,25,40,26,50,52,63)(7,18,20,44,21,45,48,65,19,13,33,34,46,36,57,59,22,37,27,53,49,61,54,67,24,35,47,58,51,60,66,68);(1,2,3,4)(5,7)(6,13)(8,24)(9,27)(10,21)(11,19)(12,22)(14,37)(15,34)(16,35)(17,36)(18,23)(20,30)(25,33)(26,51)(28,47)(29,54)(31,46)(32,49)(38,53)(39,61)(40,44)(41,58)(42,59)(43,60)(45,50)(48,56)(52,57)(55,66)(62,67)(63,65)(64,68)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,f^a=fi,c^b=ci,d^b=dfi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1263);