Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator | C22 | C22×C4×C8 |
| socle | C23 | C23×C8 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,21,18,23,30,37,19,24,17,22,29,36,31,38)(15,20,26,33,28,35,40,43,25,32,27,34,39,42,41,44);(6,9)(7,11)(13,15,17,27,18,28,31,41,19,25,16,26,29,39,30,40)(14,20,22,34,23,35,38,44,24,32,21,33,36,42,37,43);(5,7,8,11)(6,10,9,12)(15,25)(20,32)(26,27)(28,39)(33,34)(35,42)(40,41)(43,44);(6,9)(10,12)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,f^2=h,g^2=h,h^2=i,b^a=be,d^a=de,c^b=cei200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400487);