Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #386 · Out(G) = C23×D4 |
| 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 | C23 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
268 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 268 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10,13,17,14,18,12,16)(11,19,21,15,22,23,20,24);(2,6,8,7)(9,11)(10,15)(12,21)(13,20)(14,22)(16,23)(17,19)(18,24);(1,3,5,4)(2,6,8,7)(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,c^2=g,d^2=gh,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9324, #199422, #199426, … |
| maximal normal subgroup of | C2×#9324, #199422, #199426, … |
| maximal quotient of | #99964, #99966, #99988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2312);