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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(12,20)(15,22)(18,24)(21,23)(25,26)(27,32)(28,33)(29,35)(30,34)(31,36)(37,39)(38,40);(2,3)(6,7)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,27,31,33)(26,32,36,28)(29,37,30,38)(34,40,35,39);(1,3,5,2)(4,7,8,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,28)(29,37)(30,38)(31,33)(32,36)(34,39)(35,40);(1,4,5,8)(2,6,3,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28)(26,33)(27,36)(29,39)(30,40)(31,32)(34,38)(35,37);(25,29,31,30)(26,34,36,35)(27,37,33,38)(28,39,32,40);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,30,31,29)(26,35,36,34)(27,38,33,37)(28,40,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015236);