Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
449 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 449 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:
(1,2)(3,4)(9,10,13,18,14,17,12,16)(11,19,21,23,22,15,20,24);(5,6)(7,8)(9,11,14,22)(10,15,17,19)(12,21,13,20)(16,24,18,23);(2,4)(6,8)(10,16,17,18)(15,23,19,24)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94495, #94496, #133464, … |
| maximal normal subgroup of | #94495, #94496, #133464, … |
| maximal quotient of | #118673, #118723, #118727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4150);