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 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26,30,34)(27,29,33,31)(28,32,37,40)(35,36,39,38);(3,7)(6,8)(10,16)(11,12)(18,23)(19,20)(25,27,30,33)(26,29,34,31)(28,35,37,39)(32,36,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,27)(26,29)(28,35)(30,33)(31,34)(32,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,28,30,37)(26,32,34,40)(27,35,33,39)(29,36,31,38);(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,29)(26,33)(27,34)(28,36)(30,31)(32,39)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8352771);