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 | #59232 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,14)(13,20)(16,21)(17,18)(19,24)(22,23)(25,26,31,33)(27,32,30,28)(29,37,39,34)(35,36,38,40);(1,3,5,8)(2,6,4,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,31,30)(26,32,33,28)(29,35,39,38)(34,40,37,36);(10,12)(11,17)(16,20)(19,23)(26,33)(28,32)(34,37)(36,40);(1,4,5,2)(3,7,8,6)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28)(26,27)(29,36)(30,33)(31,32)(34,35)(37,38)(39,40);(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29,31,39)(26,34,33,37)(27,35,30,38)(28,36,32,40);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,30,31,27)(26,28,33,32)(29,38,39,35)(34,36,37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617000);