Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C2×#10586342 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #120 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1850 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(1,2,6,12,7,11,23,28,8,13,24,29,5,15,22,30)(3,14,17,20,10,9,25,21,18,26,32,19,16,27,31,4);(1,3,6,17,7,10,23,25,8,18,24,32,5,16,22,31)(2,9,12,21,11,26,28,19,13,27,29,4,15,14,30,20);(1,4)(2,10)(3,15)(5,19)(6,14)(7,20)(8,21)(9,23)(11,18)(12,25)(13,16)(17,30)(22,27)(24,26)(28,32)(29,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12918, #494233, #494957, … |
| maximal normal subgroup of | C2×#12918, #494233, #494957, … |
| maximal quotient of | #135819, #135831, #135836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5906);