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 | |
4408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,18)(12,17)(13,14)(15,19)(16,23)(20,22)(21,24)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,19,14)(11,20,23,21)(16,24,18,22)(25,27,31,29)(26,33,28,32)(30,35,38,37)(34,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,12)(10,17,19,14)(11,21,23,20)(16,24,18,22)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,20)(13,21)(14,24)(15,23)(16,19)(17,22)(25,29,31,27)(26,32,28,33)(30,37,38,35)(34,39,36,40);(9,14,15,17)(10,13,19,12)(11,22,23,24)(16,21,18,20)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8571629);