Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #106 |
| commutator | C8 | C2×C32 |
| Frattini, cosocle | C8×C16 | C22 |
| socle, prime radical | C22 | #1367 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C16 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(2,6,7,12)(3,9)(4,11)(8,15,14,13)(10,16)(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);(1,2,3,6,4,8,9,13,5,7,10,12,11,14,16,15)Polycyclic presentation:
2^9|a^2=d,b^2=c,c^2=e,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,d^b=dg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72227188, #72230695, #72230710, … |
| maximal normal subgroup of | #72227188, #72230695, #72230710, … |
| maximal quotient of | #72227187, #72230695 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1072);