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 | |
4411 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)(6,8)(9,10)(11,16)(12,18)(13,14)(15,17)(19,24)(20,23)(21,22)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,37,39,36);(3,7)(6,8)(9,11)(10,16)(12,14)(13,18)(15,20)(17,23)(19,22)(21,24)(25,27,30,35)(26,28,33,31)(29,34,38,39)(32,36,40,37);(1,2)(3,6)(4,5)(7,8)(10,17)(11,20)(14,19)(18,21)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(1,3)(2,6)(4,7)(5,8)(9,12)(10,18)(11,19)(13,23)(14,20)(15,22)(16,24)(17,21)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,19)(11,21)(12,16)(14,17)(15,24)(18,20)(22,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(9,14,15,19)(10,13,17,24)(11,12,20,22)(16,18,23,21)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8278520);