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 | #2669 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,12,17,16)(13,21,23,18)(19,22,24,20)(25,26)(27,33)(28,38)(29,31)(30,34)(32,35)(36,39)(37,40);(1,2)(3,8)(4,6)(5,7)(10,15)(12,16)(18,21)(20,22)(25,27,32,29)(26,33,35,31)(28,37,30,36)(34,39,38,40);(1,3)(2,4)(5,6)(7,8)(10,15)(11,17)(18,21)(19,24)(25,28,32,30)(26,34,35,38)(27,36,29,37)(31,39,33,40);(3,6)(4,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,22)(20,21)(23,24)(25,29,32,27)(26,31,35,33)(28,36,30,37)(34,40,38,39);(1,4,5,8)(2,6,7,3)(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,21,24)(25,30,32,28)(26,34,35,38)(27,37,29,36)(31,39,33,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,23)(10,18,15,21)(11,19,17,24)(12,20,16,22)(25,31)(26,27)(28,39)(29,35)(30,40)(32,33)(34,36)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068251);