Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C23×D4×#392 (order 4096) · Inn(G) = C2×D4 · Out(G) = C22×#392 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
210 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,7)(3,8,12,15,13,16,11,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(2,7,10,9)(8,14,16,15)(17,19,20,26,21,25,27,31)(18,22,24,28,23,29,30,32);(1,3,5,12,6,13,4,11)(2,8,9,15,10,16,7,14)(18,23)(19,25)(24,30)(26,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ch,d^a=dg,f^a=fg,c^b=ch76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8189, #182302, #182303, … |
| maximal normal subgroup of | C2×#8189, #182302, #182303, … |
| maximal quotient of | #96716, #96717, #96718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1177);