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 | |
4563 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,23,24,18)(25,26,29,33)(27,31,35,30)(28,32,37,36)(34,39,40,38);(2,5)(6,7)(10,15)(12,22)(16,19)(18,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(12,22)(13,24)(25,26,29,33)(27,30,35,31)(28,36,37,32)(34,39,40,38);(1,3,4,8)(2,6,5,7)(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,24)(20,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40)Polycyclic presentation:
2^9|a^2=gi,c^2=ghi,d^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900965);