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 | |
4427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26,31,30)(27,34,28,32)(29,33,38,37)(35,40,36,39);(2,3)(6,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(27,28)(32,34)(35,36)(39,40);(1,3,5,2)(4,7,8,6)(25,27,31,28)(26,32,30,34)(29,35,38,36)(33,39,37,40);(1,4,5,8)(2,6,3,7)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,28,31,27)(26,32,30,34)(29,36,38,35)(33,39,37,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29,31,38)(26,33,30,37)(27,35,28,36)(32,39,34,40);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,26)(27,34,28,32)(29,37,38,33)(35,40,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=i,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8345017);