Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12873 |
| automorphism | Aut(G) = #72314018 (order 4096) · Inn(G) = #1823 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1823 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
188 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 188 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,7,10,6,22)(3,13,15,26,16,8,21,25)(4,17,19,27,20,9,18,11)(14,23,30,29,31,28,32,24);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,23)(10,13)(11,24)(12,26)(17,28)(18,30)(19,32)(20,31)(22,25)(27,29);(2,9,10,17)(4,18)(5,6)(8,23,13,28)(11,12,27,22)(14,30)(15,21)(19,20)(24,26,29,25)(31,32)Polycyclic presentation:
2^8|a^2=f,c^2=d,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,f^b=fh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130846, #130847, #130848, … |
| maximal normal subgroup of | #130846, #130847, #130848, … |
| maximal quotient of | #135257, #135258, #135286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5862);