Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9183 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
289 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,7,5,3,4,8)(9,10)(11,17)(12,20,22,15)(13,19)(14,16)(18,21,24,23);(1,3,5,2)(4,8,6,7)(9,11,17,10,14,19,13,16)(12,18,21,15,22,24,23,20);(1,4,5,6)(2,7,3,8)(9,12)(10,15)(11,18)(13,23)(14,22)(16,20)(17,21)(19,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=h,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #132821, #132824, #132826, … |
| maximal normal subgroup of | #132821, #132824, #132826, … |
| maximal quotient of | #99833, #99834, #99847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2186);