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 | |
4193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,19,21,16)(12,22,23,17)(14,24,20,18)(25,26)(27,33)(28,38)(29,32)(30,35)(31,36)(34,39)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11,15,21)(10,16,13,19)(12,20,23,14)(17,18,22,24)(25,27,31,29)(26,32,36,33)(28,35,39,37)(30,38,40,34);(1,3)(2,5)(4,8)(6,7)(9,12,15,23)(10,17,13,22)(11,14,21,20)(16,24,19,18)(25,27,31,29)(26,33,36,32)(28,37,39,35)(30,38,40,34);(2,6)(5,7)(9,12)(10,17)(11,20)(13,22)(14,21)(15,23)(16,18)(19,24)(25,28)(26,34)(27,37)(29,35)(30,33)(31,39)(32,40)(36,38);(1,2)(3,5)(4,6)(7,8)(9,13,15,10)(11,16,21,19)(12,22,23,17)(14,24,20,18)(25,29,31,27)(26,32,36,33)(28,37,39,35)(30,38,40,34);(1,3)(2,7)(4,8)(5,6)(9,14,15,20)(10,18,13,24)(11,12,21,23)(16,22,19,17)(25,30)(26,35)(27,38)(28,33)(29,34)(31,40)(32,39)(36,37)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10126999);