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 | |
4432 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)(3,6,7,4)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,32,35)(27,36,30,33)(28,34,29,31)(37,39,38,40);(1,3)(2,6)(4,8)(5,7)(12,19)(16,22)(18,23)(20,24)(25,27,32,30)(26,33,35,36)(28,37,29,38)(31,39,34,40);(1,2)(3,4)(5,8)(6,7)(10,15)(11,17)(16,22)(18,23)(25,28,32,29)(26,34,35,31)(27,37,30,38)(33,40,36,39);(1,4)(2,7)(3,8)(5,6)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,29,32,28)(26,31,35,34)(27,38,30,37)(33,39,36,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,30,32,27)(26,33,35,36)(28,38,29,37)(31,39,34,40);(1,2)(3,6)(4,7)(5,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,31)(26,28)(27,39)(29,35)(30,40)(32,34)(33,37)(36,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=ghi,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, 9623440);