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 | #44022 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,23)(21,22)(25,26)(27,31)(28,33)(29,37)(30,32)(34,38)(35,36)(39,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,12)(10,16,17,18)(13,20,24,22)(14,21,19,23)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,39,37,36);(2,5)(6,7)(10,17)(14,19)(16,18)(21,23)(26,32)(27,33)(34,37)(35,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,11)(10,18,17,16)(13,22,24,20)(14,23,19,21)(25,28,30,31)(26,33,32,27)(29,36,38,39)(34,40,37,35);(9,13)(10,19)(11,20)(12,22)(14,17)(15,24)(16,23)(18,21)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,19)(10,13,17,24)(11,21,12,23)(16,20,18,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|b^2=h,d^2=h,f^2=g,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, 8083808);