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 | |
4503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26,30,33)(27,29,32,34)(28,31,39,37)(35,38,36,40);(3,7)(6,8)(10,11)(12,19)(16,18)(20,24)(25,27)(26,29)(28,36)(30,32)(31,40)(33,34)(35,39)(37,38);(1,2)(3,6)(4,5)(7,8)(13,21)(16,18)(20,24)(22,23)(27,32)(29,34)(35,36)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,26)(27,34)(28,37)(29,32)(30,33)(31,39)(35,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28)(26,31)(27,35)(29,38)(30,39)(32,36)(33,37)(34,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,34)(26,32,33,27)(28,38,39,40)(31,36,37,35)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325508);