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 | #52292 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4404 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,19,22,16)(12,21,23,13)(14,17,24,20)(25,26)(27,34)(28,31)(29,32)(30,33)(35,38)(36,39)(37,40);(2,4)(6,7)(9,11,15,22)(10,16,18,19)(12,20,23,17)(13,14,21,24)(26,28)(27,32)(33,36)(35,40);(1,3)(2,6)(4,7)(5,8)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,38)(39,40);(1,4,5,2)(3,7,8,6)(9,12)(10,13)(11,20)(14,16)(15,23)(17,22)(18,21)(19,24)(25,28,31,26)(27,34,32,29)(30,36,39,33)(35,38,40,37);(9,13)(10,12)(11,14)(15,21)(16,20)(17,19)(18,23)(22,24)(25,29,31,34)(26,32,28,27)(30,38,39,37)(33,35,36,40);(9,14)(10,17)(11,21)(12,16)(13,22)(15,24)(18,20)(19,23)(25,30,31,39)(26,33,28,36)(27,35,32,40)(29,37,34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003830);