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 | #38938 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4409 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26,30,31)(27,32,34,28)(29,33,38,39)(35,37,40,36);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,20)(13,21)(14,17)(15,19)(18,23)(22,24)(26,31)(27,34)(33,39)(35,40);(1,3,5,4)(2,7,6,8)(10,17)(11,19)(12,18)(21,22)(25,27)(26,32)(28,31)(29,35)(30,34)(33,37)(36,39)(38,40);(9,12,15,18)(10,13,17,24)(11,20,19,23)(14,22,16,21);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,21)(12,17)(14,20)(15,24)(16,23)(19,22)(25,28,30,32)(26,27,31,34)(29,37,38,36)(33,40,39,35);(9,14)(10,19)(11,17)(12,22)(13,23)(15,16)(18,21)(20,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7773631);