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 | |
4618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,16)(11,18,20,15)(12,17,21,24)(13,22,23,19)(25,26,30,31)(27,33,34,28)(29,37,39,32)(35,36,40,38);(1,2,4,5)(3,6,7,8)(9,11,14,20)(10,15,16,18)(12,19,21,22)(13,17,23,24)(25,27,30,34)(26,28,31,33)(29,35,39,40)(32,36,37,38);(1,3,4,7)(2,6,5,8)(10,16)(12,21)(15,18)(19,22)(26,31)(27,34)(32,37)(35,40);(9,12)(10,17)(11,19)(13,18)(14,21)(15,23)(16,24)(20,22)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,39)(38,40);(9,11)(10,18)(12,19)(13,17)(14,20)(15,16)(21,22)(23,24)(25,28)(26,27)(29,38)(30,33)(31,34)(32,40)(35,37)(36,39);(9,13)(10,19)(11,17)(12,18)(14,23)(15,21)(16,22)(20,24)(25,29)(26,32)(27,35)(28,36)(30,39)(31,37)(33,38)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7944189);