Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C24 | C22×C4 |
| Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(6,7)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(1,2)(3,4)(5,6)(7,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(2,4)(5,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=d,b^a=bd,c^a=ce,e^a=eh,c^b=cf,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97216, #97217, #97219, … |
| maximal normal subgroup of | #97216, #97217, #97219, … |
| maximal quotient of | #118695, #119451, #119457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4185);