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 | #2847 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,19)(11,16,21,15)(12,17,22,23)(13,18,24,20)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,20)(13,23)(14,21)(16,19)(17,24)(18,22)(25,27,29,35)(26,30,33,31)(28,34,38,37)(32,39,36,40);(1,3,5,4)(2,7,6,8)(9,11,14,21)(10,16,19,15)(12,18,22,20)(13,17,24,23)(25,27)(26,31)(28,37)(29,35)(30,33)(32,39)(34,38)(36,40);(9,11)(10,16)(12,20)(13,17)(14,21)(15,19)(18,22)(23,24);(1,4,5,3)(2,8,6,7)(9,12,14,22)(10,17,19,23)(11,20,21,18)(13,15,24,16)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(9,13)(10,18)(11,17)(12,15)(14,24)(16,22)(19,20)(21,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7641561);