Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #106 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #340 |
| prime radical | C23 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10,13,16,15,19,23,18)(11,14,20,24,17,12,21,22);(2,6,8,7)(9,11,14,16,13,21,24,10,15,17,12,18,23,20,22,19)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=ef,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7174, #126777, #126838, … |
| maximal normal subgroup of | C2×#7174, #126777, #126838, … |
| maximal quotient of | #92934, #92938, #92950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 162);