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 | |
4399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,15)(11,19,20,16)(12,21,22,17)(13,23,24,18)(25,26,30,34)(27,33,29,31)(28,32,36,39)(35,40,37,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(17,21)(18,23)(25,27,30,29)(26,31,34,33)(28,35,36,37)(32,38,39,40);(2,5)(6,8)(10,15)(12,22)(16,19)(18,23)(28,36)(32,39)(35,37)(38,40);(9,11,14,20)(10,16,15,19)(12,18,22,23)(13,21,24,17)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,19,24,16)(25,28,30,36)(26,32,34,39)(27,35,29,37)(31,38,33,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,21)(12,19)(14,24)(15,23)(16,22)(17,20)(25,29,30,27)(26,33,34,31)(28,37,36,35)(32,40,39,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8030102);