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 | |
1617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,13,15)(11,12,16,14)(19,23)(21,24)(25,26,30,33)(27,28,34,31)(29,37,38,32)(35,39,40,36);(1,3,5,8)(2,6,7,4)(9,11)(10,14)(12,15)(13,16)(17,18,20,22)(19,21,23,24)(27,34)(28,31)(35,40)(36,39);(2,7)(4,6)(9,11,13,16)(10,14,15,12)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(25,28,30,31)(26,27,33,34)(29,36,38,39)(32,35,37,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,29,30,38)(26,32,33,37)(27,35,34,40)(28,36,31,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fgi,d^2=fg,e^2=fg,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, 4540025);