Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = C2×#72267396 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = C23×#41 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle, prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,17,20,19,22,27,15,16,21,26,25,28,23,18,24);(2,4)(5,6,8,11,9,12,7,10)(13,15,18,23,28,20,26,21,16,24,27,22,19,25,17,14)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fg,g^2=h,b^a=bc,c^a=cf,e^a=ef,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7170, #126765, #126766, … |
| maximal normal subgroup of | C2×#7170, #126765, #126766, … |
| maximal quotient of | #92951, #92952, #92953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 158);