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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1520 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)(3,4)(5,8)(6,7)(9,10,14,12)(11,15,13,16)(19,20)(23,24)(25,26,30,31)(27,32,33,28)(29,34,38,39)(35,37,40,36);(2,7)(3,5)(9,11,14,13)(10,15,12,16)(18,22)(23,24)(26,31)(27,33)(34,39)(35,40);(1,3,6,5)(2,8,7,4)(9,12,14,10)(11,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,32)(28,31)(29,35)(30,33)(34,37)(36,39)(38,40);(1,4)(2,3)(5,7)(6,8)(17,19,21,20)(18,23,22,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,5,6,3)(2,8,7,4)(9,13,14,11)(10,15,12,16)(17,20,21,19)(18,24,22,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=ghi,c^2=hi,d^2=i,e^2=fgh,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, 5853876);