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 | |
4661 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(27,34)(28,31)(35,39)(36,37);(1,2)(3,7)(4,8)(5,6)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,27,30,34)(26,31,33,28)(29,35,38,39)(32,37,40,36);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,40)(38,39);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28)(26,27)(29,37)(30,31)(32,39)(33,34)(35,40)(36,38);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,29,30,38)(26,32,33,40)(27,35,34,39)(28,36,31,37)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=g,f^2=h,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, 7809502);