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 | |
4848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26)(27,30)(28,35)(29,31)(32,36)(33,39)(34,37)(38,40);(2,5)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27,29,34)(26,30,31,37)(28,33,36,40)(32,38,35,39);(1,2)(3,6)(4,5)(7,8)(12,20)(15,22)(18,24)(21,23)(26,31)(30,37)(32,35)(38,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,32)(29,31)(33,38)(34,37)(35,36)(39,40);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,28)(26,32)(27,33)(29,36)(30,38)(31,35)(34,40)(37,39)Polycyclic presentation:
2^9|a^2=gi,b^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9343611);