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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(6,7)(11,17)(12,20)(15,18)(16,21)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,37,39,36);(1,2,5,3)(4,6,8,7)(9,10,14,13)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,30,35)(26,31,33,28)(29,34,38,39)(32,37,40,36);(1,2,5,3)(4,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(1,2,5,3)(4,6,8,7)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(1,3,5,2)(4,7,8,6)(9,12,14,20)(10,16,13,21)(11,19,17,23)(15,22,18,24);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,15,17,18)(12,21,20,16)(19,22,23,24)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103894);