Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,19)(12,13)(14,18)(15,17)(16,22)(20,21)(23,24)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,2,5,3)(4,7,8,6)(9,11,15,22)(10,16,17,19)(12,20,18,24)(13,23,14,21)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,15,18)(10,13,17,14)(11,20,22,24)(16,23,19,21)(27,33)(28,35)(30,36)(31,38);(2,3)(6,7)(10,17)(12,18)(16,19)(20,24)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40);(1,4,5,8)(2,6,3,7)(9,13,15,14)(10,12,17,18)(11,21,22,23)(16,24,19,20)(27,33)(30,36)(34,39)(37,40);(9,14,15,13)(10,18,17,12)(11,21,22,23)(16,24,19,20)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011654);