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 | |
4567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(28,36)(32,39)(35,40)(37,38);(1,2,5,3)(4,6,8,7)(11,18)(14,15)(19,24)(21,22)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,2,5,3)(4,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(1,3,5,2)(4,7,8,6)(9,12,13,20)(10,16,17,23)(11,19,18,24)(14,21,15,22)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,16)(13,17)(15,18)(19,21)(20,23)(22,24)(25,29)(26,33)(27,34)(28,37)(30,31)(32,40)(35,39)(36,38)Polycyclic presentation:
2^9|b^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8341909);