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 | |
4365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,16)(12,13)(14,24)(15,18)(17,22)(19,23)(20,21)(28,36)(32,37)(35,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,18,24)(12,21,22,19)(13,20,17,23)(26,30)(27,33)(32,37)(35,39);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,17,18,13)(11,19,14,21)(16,20,24,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(9,13,15,17)(10,12,18,22)(11,20,14,23)(16,21,24,19);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,19)(13,18)(14,21)(15,22)(16,20)(23,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(9,14,15,11)(10,16,18,24)(12,21,22,19)(13,23,17,20)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7627183);