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 | |
4517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,13)(14,15)(16,20)(18,23)(19,24)(21,22)(25,26,29,30)(27,31,35,33)(28,36,37,32)(34,39,40,38);(2,6)(3,7)(10,15)(12,21)(16,17)(18,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2,5,6)(3,4,7,8)(26,30)(31,33)(32,36)(38,39);(9,11,14,20)(10,16,15,17)(12,18,21,24)(13,19,22,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3)(2,4)(5,7)(6,8)(9,12,14,21)(10,13,15,22)(11,18,20,24)(16,19,17,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,13)(10,12)(11,19)(14,22)(15,21)(16,18)(17,24)(20,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9703321);