Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 |
| prime radical | #51535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,13,15)(11,17,19,14)(12,16,20,22)(18,23,24,21)(25,26,30,33)(27,34,36,31)(28,32,29,38)(35,40,37,39);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,14,15,17)(12,18,20,24)(16,21,22,23)(25,27,30,36)(26,31,33,34)(28,37,29,35)(32,40,38,39);(10,15)(12,20)(14,17)(18,24)(25,27)(26,31)(28,35)(29,37)(30,36)(32,39)(33,34)(38,40);(1,3,5,4)(2,7,8,6)(9,12,13,20)(10,16,15,22)(11,18,19,24)(14,21,17,23)(25,28,30,29)(26,32,33,38)(27,35,36,37)(31,39,34,40);(10,15)(14,17)(16,22)(21,23)(25,29,30,28)(26,32,33,38)(27,35,36,37)(31,40,34,39);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614877);