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 | |
4442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,21)(12,18)(13,23)(14,19)(15,17)(16,20)(22,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,39)(37,40);(3,7)(6,8)(9,11,16,15)(10,17,20,21)(12,22,13,19)(14,23,24,18)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,38)(39,40);(1,2,5,4)(3,6,7,8)(9,12,16,13)(10,18,20,23)(11,22,15,19)(14,21,24,17)(25,27)(26,29)(28,35)(30,33)(31,34)(32,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13,16,12)(10,18,20,23)(11,22,15,19)(14,17,24,21)(25,27)(26,31)(28,35)(29,34)(30,33)(32,37)(36,39)(38,40);(1,4,5,2)(3,8,7,6)(9,14,16,24)(10,19,20,22)(11,23,15,18)(12,21,13,17)(25,28,30,38)(26,32,34,36)(27,35,33,40)(29,39,31,37);(9,15,16,11)(10,17,20,21)(12,19,13,22)(14,23,24,18)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,35,36,40)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8607456);