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 | |
4475 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,15,18)(11,13,16,21)(12,14,17,23)(19,20,22,24)(25,26,32,36)(27,29,30,33)(28,37,38,34)(31,35,40,39);(1,3,5,8)(2,6,7,4)(9,11,15,16)(10,13,18,21)(12,19,17,22)(14,20,23,24)(25,27)(26,33)(28,35)(29,36)(30,32)(31,37)(34,40)(38,39);(2,7)(3,8)(11,16)(13,21)(19,22)(20,24)(25,28,32,38)(26,34,36,37)(27,35,30,39)(29,31,33,40);(1,4,5,6)(2,3,7,8)(9,12)(10,14)(11,19)(13,20)(15,17)(16,22)(18,23)(21,24)(25,29)(26,27)(28,31)(30,36)(32,33)(34,35)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,21)(10,16,18,11)(12,20,17,24)(14,22,23,19)(25,30,32,27)(26,29,36,33)(28,35,38,39)(31,34,40,37);(1,4)(2,3)(5,6)(7,8)(9,14)(10,17)(11,20)(12,18)(13,22)(15,23)(16,24)(19,21)(25,31)(26,35)(27,37)(28,33)(29,38)(30,34)(32,40)(36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528767);