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 | C2×#560 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4175 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)(12,19)(13,21)(14,16)(15,22)(18,20)(23,24)(26,31)(29,37)(32,38)(36,40);(1,2,5,3)(4,6,8,7)(9,10,14,16)(11,17,19,12)(13,15,22,21)(18,23,24,20)(28,35)(32,38)(34,39)(36,40);(2,3)(6,7)(9,11)(10,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(2,3)(4,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(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)(11,12)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24)(25,28,30,35)(26,32,31,38)(27,34,33,39)(29,36,37,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,15,16,21)(11,18,19,24)(12,20,17,23)(25,29)(26,33)(27,31)(28,36)(30,37)(32,39)(34,38)(35,40)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=gh,e^2=hi,f^2=g,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, 9012515);