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 | |
4230 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)(11,18)(12,21)(13,16)(14,20)(15,17)(19,23)(22,24)(25,26,31,35)(27,34,30,32)(28,29,33,40)(36,39,38,37);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,22)(13,19)(15,20)(16,24)(17,18)(21,23)(25,26,31,35)(27,32,30,34)(28,29,33,40)(36,37,38,39);(2,5)(6,7)(9,12,15,13)(10,16,17,21)(11,19,20,22)(14,23,18,24)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,37,40,39);(2,5)(6,7)(9,11)(10,14)(12,19)(13,22)(15,20)(16,23)(17,18)(21,24)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,12)(10,16,17,21)(11,19,20,22)(14,24,18,23)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,38)(34,36);(1,3)(2,6)(4,8)(5,7)(9,14)(10,11)(12,23)(13,24)(15,18)(16,19)(17,20)(21,22)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,39,40,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10230145);