Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | Aut(G) = #72281230 (order 2048) · Inn(G) = H(3,ℤ/4) · Out(G) = C2×C42 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | #405 | C22 |
| socle | C2 | #1290 |
| prime radical | C23 | #108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,13,28,41,8,14,29,12,22,40,6,30)(3,15,17,42,19,44,51,64,20,45,52,43,46,63,18,53)(4,21,23,54,25,37,24,39,26,55,27,57,56,10,35,36)(9,31,32,59,33,60,58,16,34,47,38,49,61,48,62,50);(1,3,6,18,22,46,29,52,8,20,28,51,7,19,5,17)(2,9,12,32,13,33,30,58,14,34,11,38,40,61,41,62)(4,16,24,48,56,59,23,47,26,50,35,60,25,49,27,31)(10,15,36,43,21,44,54,53,37,45,39,42,55,63,57,64)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,d^2=f,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7240, #127408, #127409, … |
| maximal normal subgroup of | C2×#7240, #127408, #127409, … |
| maximal quotient of | #93328, #93355, #93571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 228);