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 | |
4621 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)(16,19)(18,21)(20,22)(23,24)(25,26,30,31)(27,33,35,28)(29,37,38,32)(34,36,40,39);(1,2)(3,7)(4,5)(6,8)(10,15)(13,16)(17,22)(21,23)(26,31)(28,33)(29,38)(34,40);(2,5)(6,7)(12,20)(17,22)(18,24)(21,23)(25,27,30,35)(26,28,31,33)(29,34,38,40)(32,36,37,39);(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,24)(21,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,18)(13,21)(14,20)(15,22)(16,23)(19,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,13,14,16)(10,11,15,19)(12,21,20,23)(17,18,22,24)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=h,c^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=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9144250);