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 | |
4597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,17)(16,21)(22,23)(25,26)(27,35)(28,36)(29,34)(30,32)(31,37)(33,39)(38,40);(1,2,5,6)(3,7,8,4)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,30,37)(26,31,32,35)(28,34,39,38)(29,36,40,33);(2,6)(4,7)(12,20)(16,21)(18,24)(22,23)(26,32)(29,40)(31,35)(33,36);(2,6)(3,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,33)(27,34)(29,31)(30,39)(32,36)(35,40)(37,38);(1,3)(2,4)(5,8)(6,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,27)(26,31)(28,38)(29,36)(30,37)(32,35)(33,40)(34,39);(1,4)(2,3)(5,7)(6,8)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,40)(26,34,32,38)(27,36,37,33)(28,31,39,35)Polycyclic presentation:
2^9|b^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295381);