Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,31,34)(27,35,30,32)(28,33,38,29)(36,39,40,37);(6,8)(11,18)(14,17)(19,24)(22,23)(25,27,31,30)(26,32,34,35)(28,36,38,40)(29,39,33,37);(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28)(26,33)(27,36)(29,34)(30,40)(31,38)(32,37)(35,39);(1,2)(3,4)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,29,31,33)(26,28,34,38)(27,37,30,39)(32,40,35,36);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,30,31,27)(26,32,34,35)(28,36,38,40)(29,37,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3244836);