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 | |
4370 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)(11,20)(12,13)(14,22)(15,19)(16,24)(17,23)(18,21)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(3,7)(6,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,18)(19,20)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,23)(10,17,19,13)(11,21,24,14)(16,18,20,22)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,17)(10,12,19,23)(11,22,24,18)(14,16,21,20)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39);(9,14,15,21)(10,18,19,22)(11,23,24,12)(13,16,17,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8059397);