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 | C2×#538 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,26,31,33)(27,35,38,32)(28,29,39,40)(30,34,37,36);(1,2)(3,5)(4,6)(7,8)(9,11,14,18)(10,15,16,13)(12,19,20,24)(17,22,23,21)(25,27,31,38)(26,32,33,35)(28,36,39,34)(29,37,40,30);(10,16)(11,18)(12,20)(21,22)(26,33)(28,39)(32,35)(34,36);(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,28)(26,29)(27,36)(30,35)(31,39)(32,37)(33,40)(34,38);(9,11)(10,13)(12,19)(14,18)(15,16)(17,21)(20,24)(22,23)(25,29)(26,28)(27,37)(30,38)(31,40)(32,36)(33,39)(34,35);(1,3,4,7)(2,5,6,8)(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,22)(23,24)(25,30,31,37)(26,34,33,36)(27,29,38,40)(28,35,39,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622944);