Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2351 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)(10,14)(11,17)(12,20)(22,24)(25,26,30,33)(27,28,35,31)(29,32,39,40)(34,36,37,38);(6,8)(12,20)(16,21)(19,23)(22,24)(25,27,30,35)(26,31,33,28)(29,34,39,37)(32,38,40,36);(5,6)(7,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27)(26,28)(29,37)(30,35)(31,33)(32,38)(34,39)(36,40);(1,2)(3,4)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28)(26,27)(29,38)(30,31)(32,37)(33,35)(34,40)(36,39);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,32)(27,34)(28,36)(30,39)(31,38)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038628);