Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #121 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #3035 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
255 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 255 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)(3,6)(7,8)(9,10,12,17)(11,18)(13,16,21,23)(14,20)(15,22)(19,24);(1,2,6,8,4,7,3,5)(9,11,12,20)(10,14)(13,22,21,19)(15,23)(16,24)(17,18);(2,7)(5,8)(9,11)(10,15)(12,20)(13,22)(14,16)(17,24)(18,23)(19,21)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9210, #197577, #197578, … |
| maximal normal subgroup of | C2×#9210, #197577, #197578, … |
| maximal quotient of | #99861, #99862, #99865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2198);