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 | #2671 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1517 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)(11,14)(12,16)(13,15)(17,18,24,22,23,28,21,27)(19,26,29,30,20,25,31,32);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,19,23,20)(18,25,28,26)(21,31,24,29)(22,32,27,30);(9,12,13,11)(10,14,15,16)(17,20,23,19)(18,26,28,25)(21,29,24,31)(22,30,27,32);(1,3,4,7)(2,6,5,8)(17,21,23,24)(18,27,28,22)(19,29,20,31)(25,32,26,30);(2,5)(6,8)(17,22,24,28,23,27,21,18)(19,30,29,25,20,32,31,26)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^b=ei,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239439, #72257256, #72257340, … |
| maximal normal subgroup of | #72239439, #72257256, #72257340, … |
| maximal quotient of | #72239348 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6312320);