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 | |
4597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(1,2)(3,7)(4,5)(6,8)(10,13)(15,17)(16,21)(22,23)(28,35)(32,38)(34,39)(36,40);(2,5)(6,7)(12,20)(16,21)(18,24)(22,23)(26,31)(27,33)(32,38)(34,39);(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,24)(21,23)(25,27)(26,29)(28,34)(30,33)(31,37)(32,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,16,13,21)(11,18,19,24)(15,22,17,23)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,17,19,15)(12,21,20,16)(18,23,24,22)(25,29)(26,27)(28,36)(30,37)(31,33)(32,34)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9144139);