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 | |
4412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,13)(12,17)(14,15)(16,18)(19,21)(20,22)(23,24)(25,26)(27,32)(28,34)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,7)(4,5)(6,8)(10,15)(12,20)(13,16)(19,24)(25,27,31,28)(26,32,33,34)(29,38,35,36)(30,39,40,37);(2,5)(6,7)(10,15)(11,18)(12,20)(21,23)(26,33)(27,28)(29,35)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,18)(10,16,15,13)(12,19,20,24)(17,23,22,21)(25,28,31,27)(26,34,33,32)(29,38,35,36)(30,39,40,37);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,17,15,22)(11,19,18,24)(13,21,16,23)(25,29,31,35)(26,30,33,40)(27,36,28,38)(32,37,34,39);(1,3)(2,6)(4,8)(5,7)(9,13,14,16)(10,18,15,11)(12,21,20,23)(17,24,22,19)(25,30)(26,35)(27,37)(28,39)(29,33)(31,40)(32,38)(34,36)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8666527);