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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,16)(12,14)(13,22)(15,17)(18,24)(19,23)(20,21)(25,26,31,36)(27,35,30,32)(28,29,34,33)(37,40,39,38);(3,7)(6,8)(9,11,15,13)(10,16,17,22)(12,21,18,19)(14,20,24,23)(25,27,31,30)(26,32,36,35)(28,37,34,39)(29,38,33,40);(1,2,5,4)(3,6,7,8)(10,17)(11,13)(12,18)(20,23)(28,34)(29,33)(37,39)(38,40);(1,3,5,7)(2,6,4,8)(9,12,15,18)(10,14,17,24)(11,19,13,21)(16,23,22,20)(25,28,31,34)(26,33,36,29)(27,37,30,39)(32,40,35,38);(1,4,5,2)(3,8,7,6)(9,13,15,11)(10,16,17,22)(12,21,18,19)(14,23,24,20)(25,29,31,33)(26,34,36,28)(27,38,30,40)(32,39,35,37);(9,14)(10,18)(11,20)(12,17)(13,23)(15,24)(16,19)(21,22)(25,30,31,27)(26,35,36,32)(28,39,34,37)(29,40,33,38)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=hi,d^2=gh,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=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034717);