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 | |
4349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(12,20)(16,21)(18,24)(22,23)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,3,5,8)(2,6,4,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(10,14)(15,17)(16,21)(22,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551049);