Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #72283979 (order 2048) · Inn(G) = C42 · Out(G) = #2010 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C42 |
| commutator | C4 | C4×C16 |
| Frattini, cosocle | C4×C16 | C22 |
| socle, prime radical | C22 | #346 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10,12,19,14,20,15,13,17,22,32,33,16,21,31,34)(11,23,24,39,26,37,40,38,29,36,30,25,28,18,27,35);(2,5)(6,8)(9,11,13,25,15,27,21,36,16,28,19,38,12,30,22,37,17,29,34,39,31,40,20,23,14,26,33,35,32,24,10,18)Polycyclic presentation:
2^8|a^2=c,b^2=d,c^2=e,d^2=f,e^2=h,f^2=g,g^2=h,b^a=be,d^a=dh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #93668, #93757, #93758, … |
| maximal normal subgroup of | #93668, #93757, #93758, … |
| maximal quotient of | #93667, #93990, #94509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 320);