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 | |
4613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26,30,34)(27,35,37,31)(28,38,36,32)(29,33,39,40);(2,3)(6,7)(10,11)(15,17)(16,18)(22,23)(25,27,30,37)(26,31,34,35)(28,29,36,39)(32,33,38,40);(1,3,5,2)(4,7,8,6)(10,11)(12,19)(16,18)(20,24)(28,36)(29,39)(32,38)(33,40);(1,4,5,8)(2,6,3,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,30,36)(26,32,34,38)(27,29,37,39)(31,33,35,40);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29)(26,33)(27,36)(28,37)(30,39)(31,38)(32,35)(34,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034810);