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 | |
4660 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)(3,7)(4,6)(5,8)(9,10,15,17)(11,18,14,16)(12,21,22,13)(19,24,23,20)(25,26,31,35)(27,36,38,32)(28,29,33,37)(30,34,39,40);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,17,18)(12,19,22,23)(13,24,21,20)(25,27,31,38)(26,32,35,36)(28,34,33,40)(29,30,37,39);(1,2)(3,5)(4,6)(7,8)(10,17)(12,22)(16,18)(19,23)(25,28,31,33)(26,29,35,37)(27,34,38,40)(30,36,39,32);(1,3)(2,5)(4,8)(6,7)(9,12,15,22)(10,13,17,21)(11,19,14,23)(16,24,18,20)(25,26)(27,32)(28,29)(30,34)(31,35)(33,37)(36,38)(39,40);(9,13)(10,12)(11,20)(14,24)(15,21)(16,23)(17,22)(18,19)(25,29)(26,33)(27,30)(28,35)(31,37)(32,40)(34,36)(38,39);(9,14,15,11)(10,18,17,16)(12,23,22,19)(13,20,21,24)(25,30)(26,34)(27,37)(28,36)(29,38)(31,39)(32,33)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10015855);