Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(11,15)(12,16)(19,23)(22,24)(25,26,30,31)(27,28,35,32)(29,37,39,33)(34,40,38,36);(2,5)(6,8)(9,10,13,14)(11,16,15,12)(18,21)(22,24)(26,31)(28,32)(33,37)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18,20,21)(19,22,23,24)(25,27)(26,32)(28,31)(29,38)(30,35)(33,36)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(25,28,30,32)(26,27,31,35)(29,36,39,40)(33,34,37,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,29,30,39)(26,33,31,37)(27,34,35,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=fgh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4524345);