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 | |
2370 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,29,32)(27,33,35,30)(28,36,38,31)(34,39,37,40);(6,8)(10,11)(13,21)(15,17)(19,24)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(5,6)(7,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38);(1,2,3,4)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,28)(26,31)(27,34)(29,38)(30,39)(32,36)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fg,d^2=gi,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, 1455147);