Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #418 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×#1490 |
| prime radical | #3584 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,13,25,11,9,14,5,24,7,3,15,17)(4,18,21,30,23,16,31,29,19,10,20,27,22,28,32,26);(1,3)(2,8)(4,16)(5,11)(6,17)(7,14)(9,13)(10,23)(12,25)(15,24)(18,22)(19,28)(20,26)(21,29)(27,31)(30,32);(1,4)(2,10)(3,16)(5,21)(6,20)(7,22)(8,23)(9,19)(11,29)(12,30)(13,28)(14,18)(15,31)(17,26)(24,27)(25,32);(4,19)(10,18)(16,28)(20,21)(22,23)(26,29)(27,30)(31,32)Polycyclic presentation:
2^9|a^2=f,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=di,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215809, #72215869, #72219678, … |
| maximal normal subgroup of | #72215809, #72215869, #72219678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388485);