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 | |
4345 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)(11,19)(12,14)(13,16)(15,17)(18,23)(20,24)(21,22)(25,26,31,30)(27,28,32,35)(29,37,38,33)(34,40,39,36);(1,2,5,4)(3,7,8,6)(9,11,15,13)(10,16,17,19)(12,20,23,22)(14,21,18,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(2,4)(6,7)(10,17)(14,18)(16,19)(21,24)(26,30)(27,32)(33,37)(34,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,18)(11,20)(13,22)(14,17)(15,23)(16,24)(19,21)(25,28,31,35)(26,32,30,27)(29,36,38,40)(33,39,37,34);(9,13,15,11)(10,19,17,16)(12,22,23,20)(14,24,18,21)(25,29,31,38)(26,33,30,37)(27,34,32,39)(28,36,35,40);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,12,17,23)(11,21,13,24)(16,20,19,22)(25,30,31,26)(27,35,32,28)(29,37,38,33)(34,40,39,36)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=gh,e^2=h,f^2=gi,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, 8102999);