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 | |
4522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(6,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,32,31)(27,35,36,30)(28,33,39,38)(29,40,37,34);(1,2,5,4)(3,7,8,6)(12,19)(16,22)(18,23)(20,24)(25,27,32,36)(26,30,31,35)(28,34,39,40)(29,33,37,38);(1,3)(2,6)(4,7)(5,8)(10,15)(11,17)(16,22)(18,23)(25,28,32,39)(26,33,31,38)(27,34,36,40)(29,30,37,35);(1,2,5,4)(3,7,8,6)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,29,32,37)(26,34,31,40)(27,33,36,38)(28,30,39,35);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,27)(28,37)(29,39)(31,36)(32,35)(33,34)(38,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,31,32,26)(27,35,36,30)(28,38,39,33)(29,40,37,34)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=g,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9773108);