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 | #51782 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,21,24,23)(25,26,29,33)(27,31,34,30)(28,36,37,32)(35,38,40,39);(1,2,5,6)(3,7,4,8)(27,34)(30,31)(35,40)(38,39);(10,15)(11,18)(17,20)(19,24)(25,27,29,34)(26,30,33,31)(28,35,37,40)(32,38,36,39);(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,37)(38,40);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,28,29,37)(26,32,33,36)(27,35,34,40)(30,38,31,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622071);