Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
785 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 785 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,13,10,14,8,11)(7,12,17,20,15,19,16,18);(6,11,14,13)(12,18,19,20);(1,2,3,4)(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,20)(14,19);(7,15)(12,19)(16,17)(18,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124650, #124661, #251671, … |
| maximal normal subgroup of | #124650, #124661, #251671, … |
| maximal quotient of | #204041, #204042, #204056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14198);