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 | #36470 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26,32,38)(27,39,40,33)(28,37,31,34)(29,35,36,30);(1,2)(3,6)(4,7)(5,8)(9,11,15,14)(10,16,17,19)(12,20,18,22)(13,21,24,23)(25,27)(26,33)(28,29)(30,37)(31,36)(32,40)(34,35)(38,39);(1,3,5,4)(2,6,8,7)(10,17)(12,18)(16,19)(20,22)(25,28,32,31)(26,34,38,37)(27,36,40,29)(30,39,35,33);(9,12,15,18)(10,13,17,24)(11,20,14,22)(16,21,19,23)(25,29,32,36)(26,35,38,30)(27,28,40,31)(33,34,39,37);(1,4,5,3)(2,7,8,6)(9,13)(10,18)(11,21)(12,17)(14,23)(15,24)(16,22)(19,20)(25,30,32,35)(26,36,38,29)(27,37,40,34)(28,33,31,39);(9,14,15,11)(10,19,17,16)(12,22,18,20)(13,23,24,21)(25,31,32,28)(26,37,38,34)(27,36,40,29)(30,39,35,33)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7616912);