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 | #49750 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,12)(13,21)(14,15)(16,23)(17,19)(18,22)(20,24)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(2,6)(3,7)(10,15)(12,19)(16,21)(20,22)(28,35)(32,38)(34,39)(36,40);(1,3)(2,4)(5,7)(6,8)(10,15)(11,17)(16,21)(18,24)(26,31)(27,33)(32,38)(34,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,27)(26,29)(28,34)(30,33)(31,37)(32,36)(35,39)(38,40);(9,12)(10,11)(13,22)(14,19)(15,17)(16,24)(18,21)(20,23)(25,28,30,35)(26,32,31,38)(27,34,33,39)(29,36,37,40);(9,13,14,23)(10,16,15,21)(11,18,17,24)(12,20,19,22)(25,29)(26,27)(28,36)(30,37)(31,33)(32,34)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,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, 9622678);