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 | |
4297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,21)(12,18)(13,24)(14,20)(15,17)(16,23)(19,22)(25,26)(27,32)(28,31)(29,37)(30,34)(33,39)(35,36)(38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,19)(13,18)(14,22)(15,23)(17,21)(20,24)(25,27)(26,32)(28,36)(29,34)(30,37)(31,35)(33,40)(38,39);(1,3,5,4)(2,7,6,8)(10,17)(12,20)(16,21)(19,24)(26,28)(27,35)(33,37)(34,38);(1,3,5,4)(2,7,6,8)(9,12,15,20)(10,18,17,14)(11,19,23,24)(13,21,22,16)(25,28,31,26)(27,36,35,32)(29,37,39,33)(30,38,40,34);(1,4,5,3)(2,8,6,7)(9,13,15,22)(10,19,17,24)(11,18,23,14)(12,21,20,16)(25,29)(26,33)(27,34)(28,37)(30,36)(31,39)(32,40)(35,38);(9,14,15,18)(10,20,17,12)(11,22,23,13)(16,24,21,19)(25,30)(26,34)(27,37)(28,38)(29,32)(31,40)(33,35)(36,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=ghi,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7861788);