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 | #47852 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,17)(12,16)(13,18)(14,19)(15,20)(21,24)(22,23)(25,26,30,32)(27,28,33,31)(29,34,38,40)(35,36,37,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,16)(14,20)(17,19)(18,23)(22,24)(25,27,30,33)(26,31,32,28)(29,37,38,35)(34,36,40,39);(11,20)(12,23)(15,17)(16,22)(26,32)(27,33)(34,40)(35,37);(1,3,5,4)(2,6,8,7)(9,12,14,23)(10,16,19,22)(11,21,20,18)(13,17,24,15)(25,28,30,31)(26,33,32,27)(29,36,38,39)(34,37,40,35);(9,11)(10,17)(12,21)(13,22)(14,20)(15,19)(16,24)(18,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,37)(38,40);(1,4,5,3)(2,7,8,6)(9,13,14,24)(10,18,19,21)(11,22,20,16)(12,17,23,15)(25,29,30,38)(26,34,32,40)(27,35,33,37)(28,36,31,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=g,f^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, 7641570);