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 | #39380 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,17,20,16)(12,13,21,22)(18,23,24,19)(25,26,30,32)(27,33,34,28)(29,37,38,31)(35,36,40,39);(2,3)(6,7)(10,15)(12,21)(16,17)(18,24)(27,34)(28,33)(29,38)(31,37);(1,3,5,2)(4,7,8,6)(25,27)(26,28)(29,35)(30,34)(31,36)(32,33)(37,39)(38,40);(1,4,5,8)(2,6,3,7)(9,11,14,20)(10,16,15,17)(12,18,21,24)(13,19,22,23)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(9,12)(10,13)(11,18)(14,21)(15,22)(16,19)(17,23)(20,24)(25,29,30,38)(26,31,32,37)(27,35,34,40)(28,36,33,39);(9,13)(10,12)(11,19)(14,22)(15,21)(16,18)(17,24)(20,23)(25,26)(27,28)(29,31)(30,32)(33,34)(35,36)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015908);