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 | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4193 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)(11,16)(12,19)(13,18)(14,17)(15,20)(21,23)(22,24)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,39)(36,40);(1,2,5,3)(4,6,8,7)(9,11)(10,13)(12,21)(14,22)(15,18)(16,20)(17,23)(19,24)(26,31)(28,38)(32,34)(35,40);(2,3)(6,7)(9,11)(10,16)(12,21)(13,20)(14,22)(15,18)(17,24)(19,23)(25,27,30,37)(26,32,31,34)(28,35,38,40)(29,36,33,39);(9,12,15,14)(10,17,20,19)(11,21,18,22)(13,23,16,24)(25,28)(26,33)(27,35)(29,31)(30,38)(32,39)(34,36)(37,40);(1,3,5,2)(4,7,8,6)(9,13,15,16)(10,18,20,11)(12,23,14,24)(17,22,19,21);(1,4)(2,6)(3,7)(5,8)(9,14,15,12)(10,19,20,17)(11,22,18,21)(13,24,16,23)(25,29,30,33)(26,28,31,38)(27,36,37,39)(32,35,34,40)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gh,e^2=ghi,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, 8002098);