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 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(25,26,31,35)(27,36,30,32)(28,34,29,33)(37,39,38,40);(3,7)(6,8)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,21)(23,24)(25,27,31,30)(26,32,35,36)(28,38,29,37)(33,40,34,39);(1,2,5,4)(3,6,7,8)(10,15)(13,16)(17,20)(21,23)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,39)(38,40);(1,3,5,7)(2,6,4,8)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,28,31,29)(26,33,35,34)(27,37,30,38)(32,39,36,40);(1,4,5,2)(3,8,7,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,29,31,28)(26,34,35,33)(27,38,30,37)(32,40,36,39);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,30,31,27)(26,32,35,36)(28,38,29,37)(33,39,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=hi,e^2=h,f^2=gi,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, 8149512);