Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C22×D8 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
766 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6,8,9)(7,10,11,12)(13,14,16,21,30,38,41,36,18,23,31,35,17,22,29,20)(15,24,25,39,32,44,43,34,27,40,42,33,26,37,28,19);(1,2)(3,4)(6,9)(10,12)(13,15,18,27)(14,19,23,34)(16,28,31,43)(17,32,30,26)(20,24,36,40)(21,37,35,44)(22,39,38,33)(25,41,42,29);(2,4)(14,20)(16,29)(17,30)(19,24)(21,22)(23,36)(25,28)(26,32)(31,41)(33,44)(34,40)(35,38)(37,39)(42,43);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406833);