Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(2,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(1,3,5,4)(2,7,6,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(10,15)(14,17)(16,23)(22,24)(27,34)(30,38)(33,35)(37,39);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(28,36)(31,40)(33,35)(37,39);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=gi,c^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547462);