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 | |
4701 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)(12,13)(14,15)(16,19)(17,22)(20,23)(21,24)(26,27)(31,33)(32,34)(38,39);(1,2)(3,4)(5,6)(7,8)(10,15)(11,19)(12,22)(21,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,6)(4,7)(10,15)(13,17)(16,18)(21,23)(29,37)(32,34)(35,40)(38,39);(9,11,14,19)(10,16,15,18)(12,20,22,24)(13,21,17,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3)(2,7)(4,6)(5,8)(9,12)(10,17)(11,20)(13,15)(14,22)(16,23)(18,21)(19,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4,5,7)(2,3,6,8)(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,20,18,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|b^2=gh,d^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8139579);