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 | |
4548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,15)(11,19,21,16)(12,20,22,17)(13,18,24,23)(26,31)(27,33)(34,38)(35,40);(2,5)(6,7)(10,15)(12,22)(16,19)(18,23)(25,26)(27,28)(29,34)(30,31)(32,33)(35,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(13,24)(16,19)(17,20)(29,37)(34,38)(35,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,16,15,19)(12,23,22,18)(13,20,24,17)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,18)(13,19)(14,22)(15,20)(16,24)(21,23)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(9,13,14,24)(10,18,15,23)(11,20,21,17)(12,19,22,16)(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8309477);