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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,29,33)(27,32,35,30)(28,36,38,31)(34,39,37,40);(1,2,4,5)(3,6,7,8)(12,19)(16,22)(18,23)(20,24)(25,27,29,35)(26,30,33,32)(28,37,38,34)(31,40,36,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,22)(18,23)(28,38)(31,36)(34,37)(39,40);(2,5)(6,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28,29,38)(26,31,33,36)(27,34,35,37)(30,39,32,40);(1,2)(3,8)(4,5)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,27)(26,32)(28,37)(29,35)(30,33)(31,39)(34,38)(36,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,26)(27,30)(28,31)(29,33)(32,35)(34,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10219619);