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 | |
4527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,14)(12,20)(13,15)(16,18)(17,22)(19,24)(21,23)(26,31)(27,34)(33,36)(35,40);(1,2,5,6)(3,7,4,8)(9,11,13,18)(10,14,15,16)(12,21,22,19)(17,24,20,23)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(10,15)(11,18)(17,20)(19,21)(26,31)(29,32)(33,36)(37,39);(9,11)(10,16)(12,19)(13,18)(14,15)(17,24)(20,23)(21,22)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,22)(14,23)(15,20)(16,24)(18,21)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,4,5,3)(2,8,6,7)(9,10)(11,16)(12,17)(13,15)(14,18)(19,24)(20,22)(21,23)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)Polycyclic presentation:
2^9|b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713921);