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 | |
4679 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)(3,4,8,6)(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,23,21)(25,26,30,31)(27,28,35,32)(29,37,39,33)(34,40,38,36);(1,3,5,8)(2,6,7,4)(9,11,14,19)(10,15,17,12)(13,18,22,23)(16,21,24,20)(26,31)(28,32)(33,37)(36,40);(1,3)(2,4)(5,8)(6,7)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,37)(38,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,21)(14,15)(16,23)(17,19)(18,24)(20,22)(25,27)(26,32)(28,31)(29,38)(30,35)(33,36)(34,39)(37,40);(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(25,28,30,32)(26,27,31,35)(29,36,39,40)(33,34,37,38);(9,13,14,22)(10,16,17,24)(11,18,19,23)(12,20,15,21)(25,29,30,39)(26,33,31,37)(27,34,35,38)(28,36,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10353680);