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 | #41281 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,11)(13,21)(15,17)(19,24)(25,26,31,32)(27,36,33,28)(29,34,30,35)(37,40,38,39);(1,3,5,8)(2,6,4,7)(26,32)(27,33)(34,35)(37,38);(1,2,5,4)(3,6,8,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,31,33)(26,28,32,36)(29,37,30,38)(34,39,35,40);(1,2,5,4)(3,6,8,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28)(26,33)(27,32)(29,39)(30,40)(31,36)(34,38)(35,37);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29,31,30)(26,34,32,35)(27,37,33,38)(28,39,36,40);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,29)(26,35,32,34)(27,38,33,37)(28,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=ghi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7553067);