Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
2386 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)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,30,35)(27,32,37,31)(28,36,39,33)(29,34,40,38);(6,8)(10,14)(12,21)(15,17)(18,24)(25,27,30,37)(26,31,35,32)(28,34,39,38)(29,33,40,36);(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,27)(26,32)(28,38)(29,36)(30,37)(31,35)(33,40)(34,39);(1,2,3,4)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28,30,39)(26,33,35,36)(27,34,37,38)(29,31,40,32);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29)(26,34)(27,36)(28,31)(30,40)(32,39)(33,37)(35,38)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=g,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1455274);