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 | |
4487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,19,21,15)(12,22,23,17)(13,18,24,20)(25,26)(27,32)(28,34)(29,37)(30,36)(31,33)(35,40)(38,39);(1,2)(3,7)(4,5)(6,8)(9,11,14,21)(10,15,16,19)(12,18,23,20)(13,17,24,22)(25,27,31,30)(26,32,33,36)(28,37,38,35)(29,39,40,34);(2,5)(6,7)(10,16)(13,24)(15,19)(17,22)(26,33)(27,30)(34,39)(35,37);(9,12,14,23)(10,17,16,22)(11,20,21,18)(13,19,24,15)(25,28)(26,34)(27,35)(29,36)(30,37)(31,38)(32,40)(33,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,17)(14,21)(16,19)(18,23)(22,24)(25,29)(26,35)(27,34)(28,36)(30,39)(31,40)(32,38)(33,37);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,17)(12,19)(14,24)(15,23)(16,20)(21,22)(25,30,31,27)(26,36,33,32)(28,37,38,35)(29,39,40,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8106842);