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 | |
4603 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,29,31)(27,33,35,30)(28,32,37,39)(34,40,36,38);(1,2)(3,7)(4,6)(5,8)(11,17)(15,21)(18,23)(22,24)(25,27,29,35)(26,30,31,33)(28,36,37,34)(32,40,39,38);(1,2,4,6)(3,5,8,7)(10,14)(15,21)(16,19)(22,24)(26,31)(28,37)(30,33)(34,36);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,26)(27,30)(28,32)(29,31)(33,35)(34,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,29,37)(26,32,31,39)(27,34,35,36)(30,38,33,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,e^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, 9152365);