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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2128 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)(9,10,14,15)(11,12,16,20)(13,21,22,17)(18,23,24,19)(25,26)(27,32)(28,35)(29,31)(30,37)(33,38)(34,40)(36,39);(6,8)(10,15)(11,16)(13,22)(19,23)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(5,6)(7,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(1,2)(3,4)(9,12)(10,16)(11,15)(13,23)(14,20)(17,18)(19,22)(21,24)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(9,13,14,22)(10,17,15,21)(11,18,16,24)(12,19,20,23)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=g,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, 2976698);