Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,13)(14,16)(18,20)(22,23)(25,26,30,31)(27,34,37,32)(28,29,38,33)(35,39,40,36);(4,8)(6,7)(9,11,14,12)(10,15,16,13)(18,20)(22,23)(26,31)(29,33)(32,34)(36,39);(2,3)(6,7)(10,16)(11,12)(17,18,21,20)(19,23,24,22)(25,27,30,37)(26,32,31,34)(28,35,38,40)(29,36,33,39);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,15,16,13)(17,19,21,24)(18,22,20,23)(25,28,30,38)(26,33,31,29)(27,35,37,40)(32,39,34,36);(1,4,5,8)(2,6,3,7)(9,13,14,15)(10,11,16,12)(17,20,21,18)(19,23,24,22)(25,29,30,33)(26,28,31,38)(27,36,37,39)(32,35,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=hi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5660869);