Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #327 |
| commutator | C2×C8 | C2×C16 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7118 |
| prime radical | C23 | #350 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
119 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 119 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(1,2,5,12,7,15,30,47,8,16,31,48,26,11,28,41,9,13,32,45,25,42,57,63,6,14,29,46,4,22,24,44)(3,17,19,50,18,49,51,64,20,40,53,62,23,10,35,36,21,39,54,60,34,38,52,43,33,55,27,58,56,37,59,61);(1,3)(2,10,13,37)(4,23)(5,27)(6,33)(7,34)(8,20)(9,21)(11,40,22,55)(12,43,45,64)(14,49,16,38)(15,17,42,39)(18,25)(19,31)(24,51)(26,56)(28,52)(29,54)(30,35)(32,53)(36,48,61,46)(41,60,44,50)(47,62,63,58)(57,59)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,c^b=cf,d^b=defi,f^b=fi,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1126);