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 | #37558 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,17)(12,14)(13,18)(15,16)(19,24)(20,21)(22,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2,5,6)(3,7,4,8)(10,16)(11,12)(18,23)(19,20)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,3,5,4)(2,7,6,8)(9,11,15,12)(10,17,16,14)(13,20,22,19)(18,21,23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,12,15,11)(10,14,16,17)(13,20,22,19)(18,21,23,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4,5,3)(2,8,6,7)(9,13,15,22)(10,18,16,23)(11,19,12,20)(14,21,17,24);(9,14)(10,11)(12,16)(13,21)(15,17)(18,19)(20,23)(22,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551257);