Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
728 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 728 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) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6,8,9)(7,10,11,12)(13,14,17,23,34,39,36,43,20,22,18,24,19,25,35,41)(15,26,27,44,31,42,33,37,16,30,28,38,29,40,32,21);(1,2)(3,4)(6,9)(10,12)(13,15,20,16)(14,21,22,37)(17,32,18,33)(19,31,34,29)(23,40,24,42)(25,44,39,38)(26,43,30,41)(27,36,28,35);(2,4)(13,16,20,15)(14,21,22,37)(17,33,18,32)(19,29,34,31)(23,40,24,42)(25,44,39,38)(26,43,30,41)(27,35,28,36);(5,7,8,11)(6,10,9,12)(14,22)(21,37)(23,24)(25,39)(26,30)(38,44)(40,42)(41,43)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=i,d^2=efhi,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406842);