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 | C2×#547 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4409 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,17,12,16)(13,14,18,24)(20,23,22,21)(25,26)(27,35)(28,31)(29,33)(30,34)(32,37)(36,38)(39,40);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23)(25,27,31,37)(26,32,28,35)(29,30,36,40)(33,39,38,34);(1,2)(3,5)(4,6)(7,8)(25,26,31,28)(27,35,37,32)(29,33,36,38)(30,34,40,39);(1,3,4,8)(2,5,6,7)(9,12,15,11)(10,17,19,16)(13,22,18,20)(14,23,24,21)(25,28,31,26)(27,32,37,35)(29,38,36,33)(30,39,40,34);(9,13)(10,14)(11,20)(12,22)(15,18)(16,21)(17,23)(19,24)(25,29)(26,33)(27,30)(28,38)(31,36)(32,39)(34,35)(37,40);(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,22)(17,20)(25,30)(26,34)(27,36)(28,39)(29,37)(31,40)(32,33)(35,38)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8004277);