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 | |
4483 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)(27,31)(28,30)(29,32)(33,35)(34,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(25,26,30,28)(27,31,35,33)(29,32,37,36)(34,39,38,40);(1,2,5,4)(3,6,8,7)(10,15)(13,22)(16,23)(20,24)(26,28)(29,37)(31,33)(34,38);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,27,30,35)(26,31,28,33)(29,38,37,34)(32,40,36,39);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,e^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, 7982512);