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 |
| prime radical | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,21)(12,18)(13,24)(14,20)(15,17)(16,23)(19,22)(26,30)(28,36)(31,33)(34,40);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,19)(13,18)(14,22)(15,23)(17,21)(20,24)(26,30)(31,33)(32,37)(38,39);(10,17)(12,20)(16,21)(19,24)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(9,12,15,20)(10,18,17,14)(11,19,23,24)(13,21,22,16)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13,15,22)(10,19,17,24)(11,18,23,14)(12,21,20,16)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3,4,7)(2,6,5,8)(9,14,15,18)(10,20,17,12)(11,22,23,13)(16,24,21,19)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=h,e^2=hi,f^2=g,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, 7626247);