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 | #2388 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,16)(11,13)(12,20)(23,24)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,21)(13,18)(15,16)(17,20)(19,23)(22,24)(28,36)(32,38)(35,39)(37,40);(1,3,5,4)(2,6,7,8)(9,11,15,13)(10,14,16,18)(12,19,20,22)(17,24,21,23)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(1,2)(3,6)(4,8)(5,7)(9,12)(10,17)(11,19)(13,22)(14,24)(15,20)(16,21)(18,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,36)(33,39)(34,40);(9,13,15,11)(10,18,16,14)(12,22,20,19)(17,23,21,24)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,4,5,3)(2,6,7,8)(9,14,15,18)(10,11,16,13)(12,23,20,24)(17,22,21,19)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8336558);