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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,15,17)(11,19,13,16)(12,22,23,18)(14,20,24,21)(25,26,31,34)(27,32,35,30)(28,29,33,39)(36,37,40,38);(3,7)(6,8)(9,11,15,13)(10,16,17,19)(12,20,23,21)(14,18,24,22)(25,27,31,35)(26,32,34,30)(28,36,33,40)(29,37,39,38);(1,2,5,4)(3,6,7,8)(10,17)(14,24)(16,19)(18,22)(27,35)(30,32)(36,40)(37,38);(1,3)(2,6)(4,8)(5,7)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,19,24,16)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,39)(35,40);(1,4,5,2)(3,8,7,6)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,22,24,18)(25,29)(26,33)(27,37)(28,34)(30,36)(31,39)(32,40)(35,38);(9,14)(10,20)(11,18)(12,19)(13,22)(15,24)(16,23)(17,21)(25,30)(26,27)(28,38)(29,36)(31,32)(33,37)(34,35)(39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8192416);