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 | |
4586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,15)(11,12,18,16)(13,21,22,17)(19,23,24,20)(25,26)(27,28)(29,37)(30,31)(32,34)(33,38)(35,39)(36,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(17,21)(19,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,39)(38,40);(2,5)(6,8)(10,15)(12,16)(17,21)(20,23)(26,31)(27,34)(33,37)(35,40);(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,21)(22,24)(25,28,30,32)(26,27,31,34)(29,36,38,39)(33,35,37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=g,f^2=ghi,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, 8030086);