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 | |
4408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(11,17)(15,18)(19,23)(22,24)(25,26)(27,32)(28,37)(29,30)(31,34)(33,40)(35,39)(36,38);(2,3)(6,7)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(25,27,31,28)(26,32,34,37)(29,38,33,35)(30,36,40,39);(1,3,5,2)(4,7,8,6)(10,14)(12,20)(15,18)(19,23)(25,28,31,27)(26,32,34,37)(29,38,33,35)(30,39,40,36);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,29,31,33)(26,30,34,40)(27,35,28,38)(32,39,37,36);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,30)(26,33)(27,36)(28,39)(29,34)(31,40)(32,35)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=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, 8946606);