Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2194 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)(9,10,14,15)(11,16,17,12)(13,21,22,18)(19,20,24,23)(25,26)(27,31)(28,30)(29,36)(32,35)(33,37)(34,39)(38,40);(6,8)(10,15)(11,17)(13,22)(20,23)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(5,6)(7,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,2)(3,4)(9,12,14,16)(10,17,15,11)(13,23,22,20)(18,19,21,24)(25,28,30,26)(27,31,35,32)(29,36,37,33)(34,38,40,39);(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=fg,e^2=f,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, 2976578);