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 | |
4597 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,39)(38,40);(1,3)(2,7)(4,5)(6,8)(11,17)(15,21)(18,23)(22,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,6,4)(3,7,8,5)(10,14)(15,21)(16,19)(22,24)(26,27)(31,33)(32,34)(38,39);(2,4)(3,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,4,6,2)(3,7,8,5)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,5,6,7)(2,3,4,8)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227604);