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 | |
4417 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)(3,6,4,7)(9,10)(11,14)(12,15)(13,18)(16,21)(17,19)(20,23)(22,24)(25,26,31,36)(27,37,40,32)(28,30,35,33)(29,34,38,39);(1,3,5,4)(2,6,8,7)(10,11)(15,17)(16,18)(22,23)(25,27,31,40)(26,32,36,37)(28,29,35,38)(30,39,33,34);(1,3)(2,7)(4,5)(6,8)(10,11)(12,19)(13,21)(22,23)(28,35)(29,38)(30,33)(34,39);(3,4)(6,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,35)(26,33,36,30)(27,38,40,29)(32,39,37,34);(1,4,5,3)(2,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,38)(26,34,36,39)(27,28,40,35)(30,32,33,37);(1,2)(3,6)(4,7)(5,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,33)(26,35,36,28)(27,39,40,34)(29,32,38,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966043);