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 | #38543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,14,16)(11,12,19,15)(13,17,22,24)(18,20,21,23);(1,2)(3,8)(4,6)(5,7)(9,11,14,19)(10,15,16,12)(13,18,22,21)(17,23,24,20)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,39)(37,40);(1,3,4,7)(2,5,6,8)(10,16)(12,15)(17,24)(20,23)(26,31)(28,38)(32,40)(34,35);(2,6)(5,8)(9,11)(10,15)(12,16)(13,21)(14,19)(17,20)(18,22)(23,24)(25,27)(26,32)(28,35)(29,33)(30,37)(31,40)(34,38)(36,39);(1,3)(2,5)(4,7)(6,8)(9,12,14,15)(10,11,16,19)(13,20,22,23)(17,18,24,21)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8490604);