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 | #36362 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(10,14)(15,17)(16,20)(23,24)(25,26)(27,33)(28,35)(29,37)(30,32)(31,38)(34,36)(39,40);(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,26,30,32)(27,33,35,28)(29,31,38,37)(34,39,40,36);(1,2,5,4)(3,6,7,8)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,39)(38,40);(9,10)(11,17)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,38)(39,40);(1,4,5,2)(3,8,7,6)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|b^2=i,c^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9129321);