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 | |
4542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,19)(11,17,12,16)(13,14,18,24)(20,23,22,21)(25,26)(27,30)(28,35)(29,31)(32,36)(33,39)(34,37)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23)(25,27,29,34)(26,30,31,37)(28,33,36,40)(32,38,35,39);(1,3)(2,5)(4,8)(6,7)(26,31)(30,37)(32,35)(38,39);(9,12,15,11)(10,17,19,16)(13,22,18,20)(14,23,24,21)(25,26)(27,30)(28,32)(29,31)(33,38)(34,37)(35,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,14)(11,20)(12,22)(15,18)(16,21)(17,23)(19,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,2)(3,5)(4,7)(6,8)(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,22)(17,20)(25,28)(26,32)(27,33)(29,36)(30,38)(31,35)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575687);