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 | #34977 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,12)(13,21)(14,18)(15,16)(17,23)(19,24)(20,22)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(2,3)(6,7)(9,11,15,14)(10,12,16,18)(13,19,23,22)(17,20,21,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,3,5,2)(4,7,8,6)(10,16)(11,14)(17,21)(19,22)(25,28,31,27)(26,32,29,34)(30,35,38,36)(33,40,37,39);(1,4)(2,6)(3,7)(5,8)(9,12,15,18)(10,11,16,14)(13,20,23,24)(17,19,21,22)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(9,13,15,23)(10,17,16,21)(11,19,14,22)(12,20,18,24);(9,14,15,11)(10,18,16,12)(13,22,23,19)(17,24,21,20)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=g,d^2=gi,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, 8104213);