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 | |
4502 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)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,29)(27,34,33,30)(28,36,38,32)(35,37,39,40);(1,2)(3,8)(4,6)(5,7)(12,19)(16,22)(18,23)(20,24)(25,27,31,33)(26,30,29,34)(28,35,38,39)(32,37,36,40);(1,3,5,6)(2,4,7,8)(10,15)(11,17)(16,22)(18,23)(25,28,31,38)(26,32,29,36)(27,35,33,39)(30,37,34,40);(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,26)(27,30,33,34)(28,36,38,32)(35,37,39,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,33)(27,29)(28,37)(31,34)(32,39)(35,36)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9588956);