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 | |
4708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,18)(12,16)(13,23)(14,24)(15,17)(19,20)(21,22)(29,37)(32,39)(34,40)(36,38);(1,2,5,4)(3,7,8,6)(11,12)(16,18)(19,21)(20,22)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,18)(13,19,24,21)(14,20,23,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,16,17,18)(13,19,24,21)(14,22,23,20)(25,28,30,31)(26,27,33,35)(29,36,37,38)(32,34,39,40);(9,13)(10,14)(11,19)(12,21)(15,24)(16,20)(17,23)(18,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(9,14)(10,13)(11,20)(12,22)(15,23)(16,19)(17,24)(18,21)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8046787);