Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2676 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,24,29,23,28,21,27)(19,25,30,26,31,22,20,32);(2,5)(3,7)(17,19)(18,25)(20,21)(22,28)(23,31)(24,30)(26,29)(27,32);(9,11,13,12)(10,14,15,16)(17,20,23,30)(18,26,28,32)(19,21,31,24)(22,27,25,29);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,21,23,24)(18,27,28,29)(19,30,31,20)(22,32,25,26);(2,5)(6,8)(17,22,24,32,23,25,21,26)(18,20,29,19,28,30,27,31)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239439, #72257608, #72257667, … |
| maximal normal subgroup of | #72239439, #72257608, #72257667, … |
| maximal quotient of | #72239393 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6313725);