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:
(1,2,5,3)(4,7,8,6)(9,10,16,15)(11,19,14,17)(12,18,13,22)(20,24,21,23)(26,30)(31,37)(32,38)(39,40);(2,3)(6,7)(9,11,16,14)(10,17,15,19)(12,20,13,21)(18,23,22,24)(26,30)(27,33)(32,38)(34,35);(1,3,5,2)(4,7,8,6)(9,12,16,13)(10,18,15,22)(11,20,14,21)(17,23,19,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,4,5,8)(2,6,3,7)(9,13,16,12)(10,18,15,22)(11,21,14,20)(17,23,19,24)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(9,14,16,11)(10,19,15,17)(12,21,13,20)(18,24,22,23);(9,15,16,10)(11,19,14,17)(12,22,13,18)(20,24,21,23)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8368420);