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 | #39813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(26,31)(27,33)(32,38)(34,39);(2,7)(3,4)(10,15)(11,20)(17,21)(18,24)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(1,3,6,4)(2,8,7,5)(10,15)(12,22)(16,19)(18,24)(28,35)(32,38)(34,39)(36,40);(1,4,6,3)(2,5,7,8)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,21,23,17)(25,27)(26,29)(28,34)(30,33)(31,37)(32,36)(35,39)(38,40);(1,5)(2,3)(4,7)(6,8)(9,12)(10,17)(11,18)(13,16)(14,22)(15,21)(19,23)(20,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40);(9,13)(10,18)(11,17)(12,16)(14,23)(15,24)(19,22)(20,21)(25,29)(26,33)(27,31)(28,36)(30,37)(32,39)(34,38)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8435319);