Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,10)(5,9)(6,8)(7,12)(13,16)(14,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(19,25)(22,28)(26,31)(29,32);(2,8)(7,15)(9,10)(11,16)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(18,23)(22,28)(24,30)(29,32)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,d^a=dgh,e^a=eg,c^b=ch,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #210201, #210203, #210205, … |
| maximal normal subgroup of | #210201, #210203, #210205, … |
| maximal quotient of | #182518, #182553, #183462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10932);