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 |
| prime radical | #34727 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,12,19,15)(13,16,22,24)(18,20,21,23)(25,26,30,31)(27,29,33,35)(28,32,38,40)(34,39,36,37);(2,5)(6,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,20)(18,22)(23,24)(25,27)(26,29)(28,36)(30,33)(31,35)(32,37)(34,38)(39,40);(1,2,4,5)(3,6,7,8)(13,22)(16,24)(18,21)(20,23)(26,31)(27,33)(32,40)(34,36);(3,7)(6,8)(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,24,21)(25,28,30,38)(26,32,31,40)(27,34,33,36)(29,39,35,37);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,17,24)(11,18,19,21)(12,20,15,23)(25,26)(27,35)(28,32)(29,33)(30,31)(34,37)(36,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,22)(23,24)(25,29,30,35)(26,33,31,27)(28,37,38,39)(32,34,40,36)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8508246);