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 | |
4295 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)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26,30,32)(27,28,33,31)(29,34,37,39)(35,36,40,38);(1,3,5,8)(2,6,4,7)(25,27,30,33)(26,31,32,28)(29,35,37,40)(34,38,39,36);(1,4,5,2)(3,7,8,6)(10,15)(13,16)(17,23)(21,24)(26,32)(27,33)(29,37)(36,38);(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,21,23,24)(25,28,30,31)(26,33,32,27)(29,38,37,36)(34,35,39,40);(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,24)(18,20)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(9,13,14,16)(10,18,15,11)(12,21,22,24)(17,20,23,19)(25,29,30,37)(26,34,32,39)(27,35,33,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=h,f^2=h,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, 7982697);