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 | |
4471 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)(9,10,14,17)(11,15,21,24)(12,16,22,13)(18,19,20,23)(25,26,31,29)(27,34,32,28)(30,37,39,33)(35,36,40,38);(3,7)(6,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,23)(17,24)(18,22)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,16,17,13)(11,18,21,20)(15,19,24,23)(25,28,31,34)(26,32,29,27)(30,38,39,36)(33,35,37,40);(9,13)(10,12)(11,19)(14,16)(15,20)(17,22)(18,24)(21,23)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,38)(39,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,17)(14,22)(15,23)(18,21)(19,24)(25,29,31,26)(27,28,32,34)(30,37,39,33)(35,36,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,23)(17,24)(18,22)(25,30,31,39)(26,33,29,37)(27,35,32,40)(28,36,34,38)Polycyclic presentation:
2^9|a^2=gi,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8851215);