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 | |
2210 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)(11,16)(12,18)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26,29,32)(27,33,35,30)(28,31,38,37)(34,40,36,39);(6,8)(10,15)(11,18)(13,22)(20,23)(25,27,29,35)(26,30,32,33)(28,36,38,34)(31,40,37,39);(5,6)(7,8)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,38)(37,40);(1,2)(3,4)(9,12,14,16)(10,11,15,18)(13,23,22,20)(17,24,21,19)(25,26)(27,33)(28,37)(29,32)(30,35)(31,38)(34,39)(36,40);(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,28,29,38)(26,31,32,37)(27,34,35,36)(30,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,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, 2976741);