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 | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26,31,30)(27,32,37,36)(28,38,33,29)(34,40,39,35);(1,2,5,4)(3,7,8,6)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27,31,37)(26,32,30,36)(28,34,33,39)(29,35,38,40);(2,4)(6,7)(9,11,14,12)(10,15,18,16)(13,20,21,19)(17,23,24,22)(25,28)(26,29)(27,34)(30,38)(31,33)(32,35)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27)(26,32)(28,34)(29,35)(30,36)(31,37)(33,39)(38,40);(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,29,31,38)(26,33,30,28)(27,35,37,40)(32,39,36,34);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,26)(27,36,37,32)(28,38,33,29)(34,40,39,35)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103089);