Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4164 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)(3,6,4,7)(9,10)(11,20)(12,23)(13,18)(14,17)(15,19)(16,22)(21,24)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,13)(14,21)(15,22)(17,18)(19,20)(23,24)(26,27)(31,33)(32,34)(38,39);(9,12,15,14)(10,17,19,23)(11,13,22,21)(16,18,20,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,4)(2,6,8,7)(9,13,15,21)(10,18,19,24)(11,12,22,14)(16,17,20,23)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,3)(2,7,8,6)(9,14,15,12)(10,17,19,23)(11,21,22,13)(16,18,20,24);(9,10)(11,16)(12,17)(13,18)(14,23)(15,19)(20,22)(21,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7563190);