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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,39)(38,40);(1,3,5,8)(2,6,4,7)(25,26,30,31)(27,28,32,35)(29,33,37,39)(34,36,38,40);(1,2,5,4)(3,6,8,7)(10,15)(13,22)(16,23)(20,24)(26,31)(28,35)(29,37)(34,38);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,27,30,32)(26,28,31,35)(29,38,37,34)(33,40,39,36);(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,38)(34,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,29,30,37)(26,33,31,39)(27,34,32,38)(28,36,35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982520);