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 | |
4413 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)(11,19)(12,23)(13,18)(14,16)(15,22)(17,21)(20,24)(25,26)(27,32)(28,31)(29,37)(30,34)(33,39)(35,36)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,27)(26,32)(28,36)(29,34)(30,37)(31,35)(33,40)(38,39);(1,3,5,4)(2,7,6,8)(10,16)(15,19)(17,23)(18,24)(26,28)(27,35)(33,37)(34,38);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28,31,26)(27,36,35,32)(29,37,39,33)(30,38,40,34);(1,4,5,3)(2,8,6,7)(9,12,14,21)(10,17,16,23)(11,20,22,13)(15,24,19,18)(25,29)(26,33)(27,34)(28,37)(30,36)(31,39)(32,40)(35,38);(9,13,14,20)(10,18,16,24)(11,21,22,12)(15,23,19,17)(25,30)(26,34)(27,37)(28,38)(29,32)(31,40)(33,35)(36,39)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,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, 7811064);