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 | |
4333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(25,26,31,33)(27,32,36,39)(28,30,35,40)(29,38,37,34);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,27)(26,32)(28,37)(29,35)(30,34)(31,36)(33,39)(38,40);(10,17)(12,22)(16,19)(20,23)(26,33)(27,36)(28,35)(34,38);(1,3,5,4)(2,6,8,7)(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,28,31,35)(26,30,33,40)(27,37,36,29)(32,34,39,38);(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,29,31,37)(26,34,33,38)(27,35,36,28)(30,39,40,32);(1,4,5,3)(2,7,8,6)(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,30)(26,35)(27,34)(28,33)(29,32)(31,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7765106);