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 | |
4474 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40);(1,3)(2,6)(4,7)(5,8)(11,17)(15,21)(18,23)(22,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(10,14)(15,21)(16,19)(22,24)(25,27,29,35)(26,31,33,30)(28,34,37,40)(32,39,36,38);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,3)(2,7)(4,6)(5,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9626489);