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 | |
4519 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)(10,11)(15,17)(16,18)(22,23)(25,26,31,33)(27,37,38,32)(28,40,39,34)(29,30,36,35);(1,2,5,6)(3,7,4,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,27)(26,32)(28,36)(29,39)(30,40)(31,38)(33,37)(34,35);(10,11)(12,19)(16,18)(20,24)(26,33)(27,38)(29,36)(34,40);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,31,39)(26,34,33,40)(27,36,38,29)(30,32,35,37);(1,4,5,3)(2,8,6,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29,31,36)(26,35,33,30)(27,39,38,28)(32,34,37,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,35)(26,36,33,29)(27,34,38,40)(28,32,39,37)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650763);