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 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,20)(21,23)(22,24)(25,26,29,33)(27,31,34,30)(28,32,37,36)(35,39,40,38);(1,2)(3,6)(4,7)(5,8)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(10,15)(13,17)(16,23)(20,24)(27,34)(30,31)(35,40)(38,39);(9,11)(10,16)(12,22)(13,20)(14,21)(15,23)(17,24)(18,19)(25,26)(27,30)(28,36)(29,33)(31,34)(32,37)(35,39)(38,40);(1,3,5,4)(2,6,8,7)(9,12,14,18)(10,17,15,13)(11,19,21,22)(16,20,23,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,4,5,3)(2,7,8,6)(9,13,14,17)(10,18,15,12)(11,20,21,24)(16,19,23,22)(25,28,29,37)(26,32,33,36)(27,35,34,40)(30,38,31,39)Polycyclic presentation:
2^9|a^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7974348);