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 | |
4624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,40)(35,38);(1,2)(3,7)(4,5)(6,8)(10,11)(13,21)(15,17)(20,24)(25,27,30,33)(26,28,31,36)(29,38,39,34)(32,40,37,35);(2,5)(6,7)(12,19)(15,17)(20,24)(22,23)(26,31)(27,33)(32,37)(34,38);(2,5)(6,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,35)(30,39)(31,37)(33,38)(36,40)Polycyclic presentation:
2^9|b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227819);