Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10,14,17)(11,18,15,12)(13,21,22,16)(19,20,23,24)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,39,37);(6,8)(11,15)(12,18)(19,23)(20,24)(25,27,30,32)(26,31,34,28)(29,35,38,39)(33,37,40,36);(5,6)(7,8)(9,11,14,15)(10,12,17,18)(13,19,22,23)(16,20,21,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,37)(36,40)(38,39);(1,2)(3,4)(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,28,30,31)(26,32,34,27)(29,37,38,36)(33,35,40,39);(9,13,14,22)(10,16,17,21)(11,19,15,23)(12,20,18,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2667895);