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 | #36727 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,16)(12,18)(13,23)(14,20)(15,17)(19,21)(22,24)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,2,5,3)(4,6,8,7)(9,11,15,20)(10,16,17,14)(12,21,22,13)(18,19,24,23)(26,30)(31,33)(32,36)(38,39);(2,3)(6,7)(10,17)(11,20)(13,21)(18,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(9,12)(10,18)(11,13)(14,19)(15,22)(16,23)(17,24)(20,21)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3,5,2)(4,7,8,6)(9,13)(10,19)(11,12)(14,18)(15,21)(16,24)(17,23)(20,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,11)(12,19)(13,18)(15,16)(17,20)(21,24)(22,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,e^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, 8027285);