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 | |
4588 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)(3,5)(4,7)(6,8)(9,10)(11,19)(12,22)(13,21)(14,15)(16,20)(17,23)(18,24)(26,27)(29,37)(31,33)(36,40);(2,3)(6,7)(10,15)(11,20)(17,22)(18,21)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,38,40,39);(1,2,5,3)(4,6,8,7)(10,15)(12,23)(16,19)(18,21)(28,35)(31,33)(36,40)(38,39);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,23)(22,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,4,5,8)(2,6,3,7)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(9,13)(10,18)(11,17)(12,19)(14,24)(15,21)(16,23)(20,22)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=gh,c^2=i,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026131);