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 | |
4470 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)(11,16)(12,18)(13,21)(14,20)(15,17)(19,24)(22,23)(25,26,31,32)(27,29,33,28)(30,34,39,40)(35,37,38,36);(1,2)(3,7)(4,5)(6,8)(9,11,15,14)(10,16,17,20)(12,21,22,19)(13,23,24,18)(26,32)(28,29)(34,40)(36,37);(2,5)(6,7)(10,17)(11,14)(18,23)(19,21)(25,27)(26,28)(29,32)(30,38)(31,33)(34,37)(35,39)(36,40);(9,12)(10,18)(11,19)(13,20)(14,21)(15,22)(16,24)(17,23)(25,28,31,29)(26,33,32,27)(30,36,39,37)(34,38,40,35);(1,3)(2,6)(4,8)(5,7)(9,13)(10,19)(11,18)(12,20)(14,23)(15,24)(16,22)(17,21)(25,29,31,28)(26,27,32,33)(30,37,39,36)(34,35,40,38);(1,2)(3,6)(4,5)(7,8)(9,14,15,11)(10,20,17,16)(12,21,22,19)(13,23,24,18)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=g,e^2=g,f^2=gh,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, 8106860);