Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| 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 | #3109 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
265 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,4,6,5,3,8,7)(9,10,13,16,14,17,12,22)(11,18,21,23,19,24,20,15);(1,3)(2,5)(4,6)(7,8)(9,11)(10,15,17,23)(12,20)(13,21)(14,19)(16,18,22,24);(1,2)(3,5)(4,6)(7,8)(11,19)(15,23)(18,24)(20,21)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=gh,e^2=h,f^2=gh,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×#9329, #199434, #199435, … |
| maximal normal subgroup of | C2×#9329, #199434, #199435, … |
| maximal quotient of | #99975, #99977, #99983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2317);