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 | #51585 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4254 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)(11,17)(12,20)(13,15)(14,19)(16,22)(18,23)(21,24)(26,31)(29,37)(32,38)(36,40);(1,2,5,3)(4,6,8,7)(9,11,13,19)(10,14,15,17)(12,18,22,21)(16,23,20,24)(28,35)(32,38)(34,39)(36,40);(2,3)(6,7)(10,15)(14,17)(16,20)(23,24)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(2,3)(4,8)(9,11)(10,14)(12,21)(13,19)(15,17)(16,24)(18,22)(20,23)(25,27,30,33)(26,29,31,37)(28,34,35,39)(32,36,38,40);(1,3,5,2)(4,7,8,6)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,28,30,35)(26,32,31,38)(27,34,33,39)(29,36,37,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,20)(17,24)(19,21)(25,29)(26,33)(27,31)(28,36)(30,37)(32,39)(34,38)(35,40)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012509);