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 | |
4563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,15,12)(11,16,17,13)(14,20,22,18)(19,21,24,23)(26,29)(27,33)(32,36)(34,39);(1,2,5,3)(4,6,8,7)(9,11,15,17)(10,16,12,13)(14,19,22,24)(18,23,20,21)(25,26,30,29)(27,31,33,35)(28,32,37,36)(34,38,39,40);(2,3)(6,7)(10,12)(11,17)(18,20)(19,24)(27,33)(31,35)(34,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,15,10)(11,16,17,13)(14,20,22,18)(19,23,24,21)(25,27,30,33)(26,31,29,35)(28,34,37,39)(32,38,36,40);(9,13,15,16)(10,17,12,11)(14,21,22,23)(18,24,20,19)(25,28,30,37)(26,32,29,36)(27,34,33,39)(31,38,35,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,29,30,26)(27,35,33,31)(28,36,37,32)(34,40,39,38)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014651);