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 | |
4418 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)(9,10,14,16)(11,18,22,15)(12,17,23,21)(13,19,24,20)(25,26,31,28)(27,34,32,29)(30,33,38,36)(35,40,39,37);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,18)(17,24)(19,23)(25,27,31,32)(26,29,28,34)(30,35,38,39)(33,37,36,40);(10,16)(13,24)(15,18)(17,21)(27,32)(29,34)(35,39)(37,40);(1,3,5,4)(2,7,6,8)(9,12,14,23)(10,17,16,21)(11,20,22,19)(13,15,24,18)(25,28,31,26)(27,29,32,34)(30,33,38,36)(35,40,39,37);(9,11)(10,18)(12,19)(13,21)(14,22)(15,16)(17,24)(20,23)(25,29)(26,32)(27,28)(30,37)(31,34)(33,39)(35,36)(38,40);(1,4,5,3)(2,8,6,7)(9,13,14,24)(10,19,16,20)(11,21,22,17)(12,15,23,18)(25,30,31,38)(26,33,28,36)(27,35,32,39)(29,37,34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7988211);