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 | #47026 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,13)(12,20,21,16)(18,23,24,22)(25,26)(27,32)(28,29)(30,34)(31,35)(33,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,13,15,17)(12,18,21,24)(16,23,20,22)(27,35)(31,32)(36,38)(37,39);(2,5)(3,7)(10,15)(12,21)(13,17)(18,24)(25,27,30,35)(26,31,34,32)(28,38,33,36)(29,39,40,37);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,23)(14,21)(15,20)(17,22)(19,24)(25,28,30,33)(26,29,34,40)(27,36,35,38)(31,37,32,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,22)(14,17)(15,19)(16,24)(18,20)(21,23)(25,27)(26,32)(28,38)(29,37)(30,35)(31,34)(33,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,16)(14,15)(17,19)(18,23)(20,21)(22,24)(25,29)(26,33)(27,37)(28,34)(30,40)(31,38)(32,36)(35,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8891039);