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 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,14)(13,17)(15,19)(18,23)(20,21)(22,24)(25,26)(27,35)(28,38)(29,32)(30,36)(31,33)(34,39)(37,40);(1,2,5,4)(3,6,7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27,31,29)(26,32,33,35)(28,36,39,37)(30,38,40,34);(2,4)(6,8)(13,21)(17,20)(18,24)(22,23)(26,33)(30,40)(32,35)(34,38);(3,7)(6,8)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,28,31,39)(26,34,33,38)(27,37,29,36)(30,32,40,35);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,29,31,27)(26,35,33,32)(28,37,39,36)(30,34,40,38);(1,4,5,2)(3,8,7,6)(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,30,31,40)(26,36,33,37)(27,38,29,34)(28,32,39,35)Polycyclic presentation:
2^9|b^2=hi,d^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8249082);