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 | |
4520 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)(11,20,21,16)(12,22,19,14)(13,18,23,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,13)(14,18)(15,21)(17,20)(19,23)(22,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,2)(3,6)(4,5)(7,8)(10,17)(11,21)(12,19)(18,24)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(2,5)(6,8)(9,12,15,19)(10,14,17,22)(11,13,21,23)(16,18,20,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,2)(3,8)(4,5)(6,7)(9,13)(10,18)(11,19)(12,21)(14,20)(15,23)(16,22)(17,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,14)(10,19)(11,18)(12,17)(13,20)(15,22)(16,23)(21,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10219835);