Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #390 |
| prime radical | C23 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
196 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10,12,15,14,18,13,17)(11,19,21,24,20,23,22,16);(1,2,3,7,4,6,8,5)(10,15,18,17)(11,20)(16,23,24,19)(21,22);(9,11,13,22,14,20,12,21)(10,16,15,19,18,24,17,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=e,d^2=gh,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cg,d^b=dh,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #137023, #137024, #137037, … |
| maximal normal subgroup of | #137023, #137024, #137037, … |
| maximal quotient of | #102002, #102004, #102010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2715);