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 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,12)(11,19,20,16)(13,17,24,22)(14,18,21,23)(25,26)(27,36)(28,29)(30,35)(31,33)(32,38)(34,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,21)(14,24)(15,20)(17,18)(22,23)(25,27,31,38)(26,32,33,36)(28,35,34,40)(29,37,39,30);(11,20)(14,21)(16,19)(18,23)(26,33)(28,34)(32,36)(35,40);(1,3,5,4)(2,6,8,7)(9,12,15,10)(11,19,20,16)(13,17,24,22)(14,23,21,18)(25,28,31,34)(26,29,33,39)(27,35,38,40)(30,32,37,36);(9,13)(10,17)(11,21)(12,22)(14,20)(15,24)(16,18)(19,23)(25,29)(26,34)(27,37)(28,33)(30,38)(31,39)(32,40)(35,36);(1,4,5,3)(2,7,8,6)(9,14,15,21)(10,18,12,23)(11,13,20,24)(16,22,19,17)(25,30,31,37)(26,35,33,40)(27,29,38,39)(28,32,34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7719282);