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 | |
4556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(18,23)(19,21)(25,26)(27,34)(28,38)(29,33)(30,35)(31,32)(36,37)(39,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(26,32)(28,35)(33,34)(36,39);(1,3,5,6)(2,4,7,8)(25,27,31,29)(26,33,32,34)(28,36,35,39)(30,37,38,40);(1,2)(3,4)(5,7)(6,8)(9,11,14,17)(10,16,15,12)(13,21,22,19)(18,20,23,24)(25,28)(26,30)(27,36)(29,39)(31,35)(32,38)(33,37)(34,40);(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,29,31,27)(26,34,32,33)(28,39,35,36)(30,40,38,37);(1,4)(2,6)(3,7)(5,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,24)(17,21)(25,30)(26,35)(27,37)(28,32)(29,40)(31,38)(33,39)(34,36)Polycyclic presentation:
2^9|c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8331454);