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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4469 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)(11,18)(12,13)(14,15)(16,19)(17,22)(20,23)(21,24)(25,26,30,32)(27,31,35,40)(28,33,37,38)(29,36,39,34);(1,2)(3,4)(5,6)(7,8)(10,15)(11,19)(12,22)(21,23)(25,27)(26,31)(28,36)(29,38)(30,35)(32,40)(33,39)(34,37);(2,6)(4,7)(10,15)(13,17)(16,18)(21,23)(26,32)(27,35)(33,38)(34,36);(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,21,17,23)(25,28)(26,33)(27,34)(29,31)(30,37)(32,38)(35,36)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,17)(11,20)(13,15)(14,22)(16,23)(18,21)(19,24)(25,29)(26,34)(27,33)(28,31)(30,39)(32,36)(35,38)(37,40);(1,4,5,7)(2,3,6,8)(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,20,18,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,40)(36,37)(38,39)Polycyclic presentation:
2^9|a^2=gh,d^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8139514);