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 | #52375 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,6,4)(9,10,15,18)(11,19,12,16)(13,14,17,24)(20,23,22,21)(25,26,30,34)(27,35,37,31)(28,38,36,32)(29,33,40,39);(1,3)(2,4)(5,6)(7,8)(11,12)(16,19)(20,22)(21,23)(25,27,30,37)(26,31,34,35)(28,29,36,40)(32,33,38,39);(9,11,15,12)(10,16,18,19)(13,20,17,22)(14,21,24,23)(25,28,30,36)(26,32,34,38)(27,29,37,40)(31,33,35,39);(1,4,5,8)(2,6,7,3)(9,12,15,11)(10,16,18,19)(13,22,17,20)(14,21,24,23)(25,26)(27,35)(28,32)(29,39)(30,34)(31,37)(33,40)(36,38);(1,3)(2,4)(5,6)(7,8)(9,13,15,17)(10,14,18,24)(11,20,12,22)(16,21,19,23);(1,4)(2,3)(5,8)(6,7)(9,14)(10,17)(11,21)(12,23)(13,18)(15,24)(16,22)(19,20)(25,29)(26,33)(27,36)(28,37)(30,40)(31,38)(32,35)(34,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8353166);