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 | C2×#546 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,21)(20,22)(23,24)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(2,5)(3,7)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,23,22,18)(28,36)(32,39)(35,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,10,15,12)(11,19,21,16)(13,17,20,22)(14,18,24,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,23,24,18)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(9,13)(10,17)(11,14)(12,22)(15,20)(16,23)(18,19)(21,24)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,17)(19,22)(25,29)(26,33)(27,34)(28,37)(30,31)(32,40)(35,39)(36,38)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=g,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, 8004121);