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 |
| prime radical | #43526 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,16)(12,13)(14,22)(15,17)(18,24)(19,21)(20,23)(26,31)(27,29)(28,36)(39,40);(1,2,5,3)(4,6,8,7)(9,11,15,20)(10,16,17,23)(12,14,18,21)(13,22,24,19)(25,26)(27,32)(28,37)(29,34)(30,31)(33,36)(35,40)(38,39);(2,3)(6,7)(10,17)(11,20)(12,18)(19,22)(25,27,30,29)(26,32,31,34)(28,38,36,35)(33,40,37,39);(1,3,5,2)(4,7,8,6)(9,12,15,18)(10,13,17,24)(11,21,20,14)(16,19,23,22)(25,28,30,36)(26,33,31,37)(27,35,29,38)(32,39,34,40);(9,13)(10,18)(11,22)(12,17)(14,23)(15,24)(16,21)(19,20)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,21)(10,19,17,22)(11,18,20,12)(13,23,24,16)(25,29,30,27)(26,34,31,32)(28,38,36,35)(33,40,37,39)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=gi,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=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014758);