Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #386 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
229 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(2,5)(3,7)(6,8)(9,10,13,17,14,18,12,16)(11,19,21,15,22,23,20,24);(1,2,3,5,4,6,7,8)(9,11)(10,15)(12,21)(13,20)(14,22)(16,23)(17,19)(18,24);(2,6)(5,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,23)(18,24)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9319, #199332, #199336, … |
| maximal normal subgroup of | C2×#9319, #199332, #199336, … |
| maximal quotient of | #99971, #99972, #99987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2307);