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 | #46225 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,7,5)(9,10,14,15)(11,17,19,12)(13,21,22,16)(18,20,24,23)(26,31)(29,39)(32,38)(33,40);(2,4)(3,7)(10,15)(12,17)(16,21)(20,23)(27,35)(28,37)(32,38)(33,40);(1,3)(2,5)(4,8)(6,7)(25,26,30,31)(27,32,35,38)(28,33,37,40)(29,34,39,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,27)(26,32)(28,36)(29,33)(30,35)(31,38)(34,37)(39,40);(1,4,6,2)(3,8,7,5)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,28,30,37)(26,33,31,40)(27,36,35,34)(29,38,39,32);(1,5,6,8)(2,7,4,3)(9,13,14,22)(10,16,15,21)(11,18,19,24)(12,20,17,23)(25,29)(26,34)(27,33)(28,38)(30,39)(31,36)(32,37)(35,40)Polycyclic presentation:
2^9|a^2=g,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000422);