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 | #2408 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,23,24)(25,26,31,35)(27,36,37,32)(28,33,38,40)(29,30,34,39);(2,6)(4,7)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,17,22,24)(27,37)(28,38)(32,36)(33,40);(1,3)(2,7)(4,6)(5,8)(25,27)(26,32)(28,34)(29,38)(30,33)(31,37)(35,36)(39,40);(1,4,5,7)(2,3,6,8)(9,12)(10,16)(11,13)(14,20)(15,17)(18,22)(19,24)(21,23)(25,28,31,38)(26,33,35,40)(27,34,37,29)(30,36,39,32);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,29,31,34)(26,30,35,39)(27,38,37,28)(32,33,36,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,20)(12,21)(14,23)(15,22)(16,19)(18,24)(25,30)(26,34)(27,33)(28,32)(29,35)(31,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,d^2=h,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, 8353032);