Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
2216 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)(11,12,18,15)(13,21,22,16)(19,23,24,20)(26,27)(28,35)(32,34)(36,37);(6,8)(11,18)(12,15)(19,24)(20,23)(25,26,30,27)(28,33,35,31)(29,34,38,32)(36,39,37,40);(5,6,7,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,23)(20,21)(22,24)(25,27,30,26)(28,31,35,33)(29,34,38,32)(36,39,37,40);(1,2)(3,4)(9,12,14,15)(10,11,17,18)(13,20,22,23)(16,19,21,24)(25,28,30,35)(26,31,27,33)(29,37,38,36)(32,40,34,39);(9,13,14,22)(10,16,17,21)(11,19,18,24)(12,20,15,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=gh,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2978776);