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 | #42518 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(3,7)(9,10,15,20)(11,16,22,23)(12,19,14,17)(13,18,24,21)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,2,5,4)(3,6,7,8)(9,11,15,22)(10,16,20,23)(12,18,14,21)(13,19,24,17)(26,31)(27,32)(33,38)(34,39);(3,7)(6,8)(9,12,15,14)(10,17,20,19)(11,21,22,18)(13,16,24,23)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(1,3,5,7)(2,6,4,8)(9,13,15,24)(10,18,20,21)(11,17,22,19)(12,23,14,16)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(9,14,15,12)(10,19,20,17)(11,21,22,18)(13,16,24,23);(1,4,5,2)(3,8,7,6)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,24)(22,23)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=i,d^2=gi,e^2=h,f^2=g,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, 8014794);