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 | |
4587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,8,4,7)(9,10)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26)(27,30)(28,33)(29,34)(31,36)(32,38)(35,39)(37,40);(2,6)(3,4)(9,11,15,22)(10,16,18,19)(12,20,17,24)(13,21,14,23)(25,27,30,26)(28,33,36,31)(29,32,38,34)(35,39,37,40);(1,3,5,4)(2,7,6,8)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,4,5,3)(2,8,6,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40);(2,6)(7,8)(11,22)(16,19)(20,24)(21,23)(26,27)(31,33)(32,34)(39,40);(9,14,15,13)(10,17,18,12)(11,23,22,21)(16,24,19,20)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^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, 8009178);