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 | #2619 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,12)(11,19,20,16)(13,17,23,22)(14,21,24,18)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,20)(10,16,12,19)(13,21,23,18)(14,22,24,17);(11,20)(13,23)(16,19)(17,22)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,3,5,4)(2,6,8,7)(9,12,15,10)(11,19,20,16)(13,22,23,17)(14,21,24,18)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,3)(2,7,8,6)(9,13,15,23)(10,17,12,22)(11,21,20,18)(14,19,24,16)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(9,14,15,24)(10,18,12,21)(11,22,20,17)(13,19,23,16)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7550685);