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 | |
4369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,12)(13,21)(14,15)(16,23)(17,19)(18,22)(20,24)(25,26,30,32)(27,31,36,33)(28,34,39,37)(29,38,40,35);(1,3)(2,6)(4,8)(5,7)(10,15)(12,19)(16,21)(20,22)(25,27)(26,31)(28,38)(29,37)(30,36)(32,33)(34,40)(35,39);(10,15)(11,17)(16,21)(18,24)(26,32)(27,36)(34,37)(35,38);(9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,27)(26,33)(28,35)(29,37)(30,36)(31,32)(34,40)(38,39);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,22)(14,19)(15,17)(16,24)(18,21)(20,23)(25,28,30,39)(26,34,32,37)(27,35,36,38)(29,33,40,31);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,16,15,21)(11,18,17,24)(12,20,19,22)(25,29,30,40)(26,35,32,38)(27,37,36,34)(28,31,39,33)Polycyclic presentation:
2^9|a^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9622662);