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 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,19)(12,14)(13,16)(15,18)(17,23)(20,24)(21,22)(25,26)(27,34)(28,29)(30,33)(31,35)(32,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(25,27)(26,32)(28,39)(29,37)(30,36)(31,38)(33,40)(34,35);(9,11,15,13)(10,16,18,19)(12,20,23,22)(14,21,17,24)(25,28,31,30)(26,29,35,33)(27,36,38,39)(32,40,34,37);(9,12)(10,17)(11,20)(13,22)(14,18)(15,23)(16,24)(19,21)(25,29,31,33)(26,28,35,30)(27,37,38,40)(32,39,34,36);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,16,18,19)(12,22,23,20)(14,21,17,24)(25,30,31,28)(26,33,35,29)(27,36,38,39)(32,40,34,37);(9,14,15,17)(10,12,18,23)(11,21,13,24)(16,20,19,22)(25,27)(26,34)(28,36)(29,37)(30,39)(31,38)(32,35)(33,40)Polycyclic presentation:
2^9|c^2=hi,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562716);