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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,16)(13,19)(14,17)(18,21)(20,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(2,3)(6,7)(9,11,14,19)(10,15,17,13)(12,18,22,20)(16,21,24,23)(26,31)(32,33)(34,38)(39,40);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,18)(14,19)(15,17)(16,23)(20,22)(21,24)(25,27,30,28)(26,32,31,33)(29,35,37,36)(34,39,38,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,13)(12,20)(14,19)(15,17)(16,21)(18,22)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(9,12,14,22)(10,16,17,24)(11,18,19,20)(13,23,15,21)(25,28,30,27)(26,33,31,32)(29,35,37,36)(34,39,38,40);(9,13)(10,11)(12,21)(14,15)(16,20)(17,19)(18,24)(22,23)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10411807);