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 | |
4512 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(25,26)(27,33)(28,34)(29,32)(30,31)(35,39)(36,40)(37,38);(2,4)(7,8)(10,15)(12,16)(17,21)(20,23)(26,31)(27,34)(32,38)(35,40);(1,3,6,5)(2,7,4,8)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,40)(38,39);(1,2,6,4)(3,7,5,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,40)(38,39);(1,4,6,2)(3,8,5,7)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,5,6,3)(2,8,4,7)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,29,30,37)(26,32,31,38)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^9|c^2=hi,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017698);