Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C42 |
| commutator | C4 | C4×C32 |
| Frattini, cosocle | C8×C16 | C22 |
| socle, prime radical | C22 | #1312 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,3,7,4,9,11,15,5,8,10,16,12,6,13,14);(2,6,8,9)(3,10)(7,15,16,14)(11,13)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48)Polycyclic presentation:
2^9|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,h^2=i,b^a=be,d^a=dg,c^b=cg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225137, #72225140, #72230710, … |
| maximal normal subgroup of | #72225137, #72225140, #72230710, … |
| maximal quotient of | #72225140, #72230689, #72234299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 863);