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 | |
4474 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)(11,20)(12,23)(13,24)(14,19)(15,17)(16,21)(18,22)(25,26)(27,35)(28,39)(29,34)(30,32)(31,38)(33,37)(36,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,22)(13,19)(14,18)(15,21)(17,20)(23,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(1,3,5,4)(2,7,6,8)(10,17)(11,21)(12,19)(18,24)(26,32)(31,35)(33,39)(34,40);(1,3,5,4)(2,7,6,8)(9,12,15,19)(10,14,17,23)(11,22,21,13)(16,18,20,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(1,4,5,3)(2,8,6,7)(9,13,15,22)(10,18,17,24)(11,19,21,12)(14,20,23,16)(25,28,30,37)(26,33,32,39)(27,36,38,29)(31,40,35,34);(9,14)(10,19)(11,18)(12,17)(13,16)(15,23)(20,22)(21,24)(25,29,30,36)(26,34,32,40)(27,37,38,28)(31,39,35,33)Polycyclic presentation:
2^9|c^2=gh,d^2=ghi,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, 7810020);