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 | |
2368 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)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(6,8)(10,11)(12,19)(13,21)(22,23)(26,31)(27,33)(29,37)(36,39);(5,6)(7,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,2)(3,4)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=fg,c^2=g,d^2=f,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, 1512605);