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 | |
4274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,17)(11,13,16,18)(12,15,23,21)(19,22,20,24)(25,26,30,33)(27,34,31,28)(29,32,38,40)(35,39,37,36);(1,2,4,6)(3,5,8,7)(9,11)(10,13)(12,20)(14,16)(15,24)(17,18)(19,23)(21,22)(27,31)(28,34)(35,37)(36,39);(1,2)(3,5)(4,6)(7,8)(11,16)(13,18)(19,20)(22,24)(25,27,30,31)(26,28,33,34)(29,37,38,35)(32,39,40,36);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,21)(13,16)(14,17)(15,23)(19,22)(20,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(9,12)(10,15)(11,19)(13,22)(14,23)(16,20)(17,21)(18,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,37)(35,40)(38,39);(9,13,14,18)(10,16,17,11)(12,22,23,24)(15,20,21,19)(25,29,30,38)(26,32,33,40)(27,35,31,37)(28,36,34,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325567);