Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2432 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)(11,18,16,12)(13,21,24,17)(19,20,22,23)(25,26)(27,31)(28,30)(29,36)(32,35)(33,37)(34,39)(38,40);(6,8)(10,15)(11,16)(17,21)(19,22)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(5,6)(7,8)(9,11,14,16)(10,12,15,18)(13,22,24,19)(17,23,21,20)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,2)(3,4)(9,12)(10,16)(11,15)(13,23)(14,18)(17,19)(20,24)(21,22)(25,28,30,26)(27,31,35,32)(29,36,37,33)(34,38,40,39);(9,13)(10,17)(11,19)(12,20)(14,24)(15,21)(16,22)(18,23)(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=fg,c^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3240248);