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 | |
4864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,39)(35,36)(38,40);(2,5)(6,7)(9,11,13,19)(10,14,17,15)(12,18,20,24)(16,21,23,22)(25,27,30,31)(26,28,33,34)(29,35,39,38)(32,36,37,40);(1,2)(3,6)(4,5)(7,8)(12,20)(16,23)(18,24)(21,22)(25,27,30,31)(26,28,33,34)(29,38,39,35)(32,40,37,36);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,31)(27,33)(29,36)(30,34)(32,38)(35,37)(39,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,32)(27,35)(28,36)(30,39)(31,38)(33,37)(34,40)Polycyclic presentation:
2^9|b^2=gi,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833420);