Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(10,14)(12,16)(17,18,20,22)(19,21,23,24)(27,33)(28,35)(30,36)(31,38);(2,5)(3,8)(9,10,13,14)(11,12,15,16)(19,23)(21,24)(27,33)(30,36)(34,39)(37,40);(3,8)(6,7)(10,14)(11,15)(17,18)(19,24)(20,22)(21,23)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,15)(14,16)(17,19,20,23)(18,21,22,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fhi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
| maximal subgroup of | #72268312 |
| maximal normal subgroup of | #72268312 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5926645);