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 | |
4263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(1,3)(2,7)(4,5)(6,8)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,26,31,28)(27,32,29,34)(30,33,38,36)(35,39,37,40);(1,2,6,4)(3,7,8,5)(9,11,13,19)(10,14,16,17)(12,18,21,20)(15,22,24,23)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(2,4)(3,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,21)(23,24)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(1,4,6,2)(3,7,8,5)(9,12)(10,15)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(1,5,6,7)(2,3,4,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227628);