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 | |
4506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,17)(11,15,20,23)(12,13,16,21)(18,19,22,24)(26,31)(27,33)(28,35)(37,40);(3,7)(6,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,26,30,31)(27,29,33,38)(28,32,35,39)(34,37,36,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,17,23)(12,18,16,22)(13,19,21,24)(25,27,30,33)(26,29,31,38)(28,36,35,34)(32,40,39,37);(1,3)(2,6)(4,7)(5,8)(9,12)(10,13)(11,18)(14,16)(15,19)(17,21)(20,22)(23,24)(25,26)(27,29)(28,32)(30,31)(33,38)(34,37)(35,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,28,30,35)(26,32,31,39)(27,34,33,36)(29,37,38,40);(9,13)(10,16)(11,19)(12,17)(14,21)(15,22)(18,23)(20,24)(25,29)(26,33)(27,31)(28,37)(30,38)(32,36)(34,39)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8434699);