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 | |
4553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,18)(13,22)(14,20)(15,17)(19,23)(21,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,18)(13,20,23,21)(14,19,24,22)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(10,17)(11,12)(19,22)(20,21)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2,5,4)(3,6,8,7)(9,12,15,11)(10,18,17,16)(13,21,23,20)(14,22,24,19)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,2,5,4)(3,6,8,7)(9,13,15,23)(10,19,17,22)(11,20,12,21)(14,16,24,18)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,4,5,2)(3,7,8,6)(9,14)(10,20)(11,19)(12,22)(13,18)(15,24)(16,23)(17,21)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7769848);