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 | #45482 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,15,20)(11,13,18,16)(12,19,14,17)(21,24,22,23)(25,26,30,31)(27,35,38,32)(28,33,29,34)(36,40,37,39);(1,2,5,3)(4,6,8,7)(9,11)(10,16)(12,21)(13,20)(14,22)(15,18)(17,24)(19,23)(26,31)(32,35)(33,34)(39,40);(2,3)(6,7)(9,11)(10,13)(12,21)(14,22)(15,18)(16,20)(17,23)(19,24)(25,27,30,38)(26,32,31,35)(28,36,29,37)(33,39,34,40);(9,12,15,14)(10,17,20,19)(11,21,18,22)(13,23,16,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,3,5,2)(4,7,8,6)(9,13)(10,18)(11,20)(12,23)(14,24)(15,16)(17,22)(19,21)(25,28,30,29)(26,33,31,34)(27,36,38,37)(32,39,35,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,12)(10,19,20,17)(11,22,18,21)(13,24,16,23)(25,29,30,28)(26,34,31,33)(27,37,38,36)(32,40,35,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002092);