Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1926 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9546 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 64 points:
(1,2,7,14,8,15,6,31,9,16,34,13,5,12,18,36)(3,17,22,49,20,41,21,38,23,40,35,48,30,10,32,39)(4,24,27,11,26,44,33,43,29,57,60,45,28,47,50,46)(19,51,53,42,25,58,55,64,54,56,61,62,52,63,59,37);(1,3,18,21,5,30,34,22,9,23,6,32,8,20,7,35)(2,10,36,49,12,40,13,39,16,41,31,48,15,17,14,38)(4,25,33,53,28,19,27,59,29,52,50,61,26,54,60,55)(11,42,57,51,46,37,44,63,45,62,24,56,43,64,47,58);(1,4)(2,11,16,45)(3,19,23,54)(5,28)(6,33)(7,27)(8,26)(9,29)(10,37)(12,46,15,43)(13,47,14,44)(17,42)(18,50)(20,25,30,52)(21,55,32,59)(22,53,35,61)(24,31,57,36)(34,60)(38,56)(39,51)(40,62)(41,64)(48,63)(49,58)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=dg,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=egi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47344);