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 |
| prime radical | #50725 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,16)(12,18)(13,19)(14,21)(15,17)(20,24)(22,23)(25,26,30,32)(27,33,34,28)(29,31,37,38)(35,39,40,36);(1,3,6,8)(2,7,4,5)(9,11,15,13)(10,16,17,19)(12,20,22,21)(14,18,24,23)(25,26)(27,33)(28,34)(29,31)(30,32)(35,39)(36,40)(37,38);(1,2)(3,7)(4,6)(5,8)(10,17)(11,13)(12,22)(14,24)(27,34)(28,33)(29,37)(31,38);(2,4)(3,8)(9,12,15,22)(10,18,17,23)(11,21,13,20)(14,19,24,16)(25,27,30,34)(26,28,32,33)(29,35,37,40)(31,36,38,39);(1,4,6,2)(3,5,8,7)(9,13,15,11)(10,19,17,16)(12,21,22,20)(14,23,24,18)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,5)(2,3)(4,8)(6,7)(9,14)(10,20)(11,18)(12,19)(13,23)(15,24)(16,22)(17,21)(25,29)(26,31)(27,35)(28,36)(30,37)(32,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gh,e^2=h,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, 8966373);