Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #329 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7102 |
| prime radical | #532 | C8 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 157 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:
(1,2,4,10,6,13,23,39,5,11,22,37,24,40,52,61,7,12,21,38,27,41,51,62,25,9,33,34,54,35,58,59)(3,14,15,43,17,44,46,63,18,31,20,49,48,42,50,64,19,30,45,36,26,55,56,60,16,8,28,29,47,32,53,57);(1,3)(2,8)(4,20)(5,18)(6,26)(7,19)(9,30)(10,36)(11,14)(12,31)(13,42)(15,33)(16,25)(17,27)(21,28)(22,45)(23,53)(24,47)(29,37)(32,41)(34,49)(35,44)(38,43)(39,63)(40,55)(46,52)(48,54)(50,51)(56,58)(57,59)(60,62)(61,64)Polycyclic presentation:
2^9|a^2=d,c^2=f,d^2=g,e^2=i,g^2=h,h^2=i,b^a=bc,c^a=ce,f^a=fi,c^b=cf,d^b=defi,e^b=ei,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, 1084);