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 | |
4402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(12,19)(16,21)(18,23)(22,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,40)(36,39)(37,38);(3,4)(7,8)(10,14)(11,17)(16,21)(18,23)(26,31)(32,38)(33,34)(39,40);(1,2)(3,7)(4,8)(5,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,37)(26,32,31,38)(28,35,29,36)(33,39,34,40);(1,3,5,4)(2,7,6,8)(25,28,30,29)(26,33,31,34)(27,35,37,36)(32,39,38,40);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2)(3,7)(4,8)(5,6)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,28)(26,34,31,33)(27,36,37,35)(32,40,38,39)Polycyclic presentation:
2^9|c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000437);