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 | |
4539 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)(9,10,15,17)(11,20,22,16)(12,23,18,13)(14,24,21,19)(25,26,30,31)(27,28,33,36)(29,32,37,38)(34,35,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,17,20)(12,24,18,19)(13,21,23,14)(26,31)(27,33)(32,38)(34,39);(2,5)(6,8)(10,17)(12,18)(16,20)(19,24)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,18)(10,13,17,23)(11,19,22,24)(14,20,21,16)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(9,13)(10,18)(11,21)(12,17)(14,22)(15,23)(16,19)(20,24);(1,3)(2,6)(4,7)(5,8)(9,14,15,21)(10,19,17,24)(11,13,22,23)(12,20,18,16)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013250);