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 | |
4582 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)(9,10,15,16)(11,17,12,13)(14,18,23,24)(19,22,20,21)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(2,4)(6,7)(10,16)(11,12)(18,24)(19,20)(28,36)(32,39)(35,40)(37,38);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,13,16,17)(14,19,23,20)(18,21,24,22)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,16,13)(14,20,23,19)(18,22,24,21)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(9,13,15,17)(10,11,16,12)(14,22,23,21)(18,20,24,19)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(9,14,15,23)(10,18,16,24)(11,19,12,20)(13,21,17,22)(25,29)(26,33)(27,34)(28,37)(30,31)(32,40)(35,39)(36,38)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8004116);