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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1910 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(4,8)(6,7)(9,11,14,12)(10,15,16,13)(17,18)(19,23)(20,22)(21,24)(26,30)(31,33)(32,38)(39,40);(2,3)(6,7)(10,16)(11,12)(17,18,20,22)(19,21,24,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,15,16,13)(17,18)(19,21)(20,22)(23,24)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(1,4)(2,6)(3,7)(5,8)(9,13,14,15)(10,11,16,12)(17,19,20,24)(18,21,22,23)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|b^2=fghi,c^2=gi,d^2=fi,e^2=h,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, 5815597);