Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,13,16)(11,14,12,15)(25,26,30,33)(27,34,38,40)(28,32,37,39)(29,36,31,35);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(17,18,20,22)(19,23,24,21)(25,27,31,37,30,38,29,28)(26,32,36,40,33,39,35,34);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,16,14)(17,19,20,24)(18,21,22,23)(26,33)(27,37)(28,38)(29,31)(32,34)(39,40);(3,8)(6,7)(9,12,13,11)(10,15,16,14)(25,28,31,27,30,37,29,38)(26,34,36,32,33,40,35,39)Polycyclic presentation:
2^9|a^2=f,b^2=efgh,c^2=efg,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381925);