Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
2357 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)(11,12,18,16)(13,17,22,23)(19,20,24,21)(25,26)(27,31)(28,30)(29,36)(32,35)(33,37)(34,39)(38,40);(6,8)(10,15)(11,18)(17,23)(19,24)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(5,6)(7,8)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,21,23,20)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,2)(3,4)(9,12,14,16)(10,11,15,18)(13,21,22,20)(17,24,23,19)(25,28,30,26)(27,31,35,32)(29,36,37,33)(34,38,40,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(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=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,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, 2667688);