Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C24×#2397 (order 2048) · Inn(G) = C23 · Out(G) = C2×#2397 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4×C8 | C23 |
| commutator, socle | C22 | C2×C4×C8 |
| prime radical | C23 | C4×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8);(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(2,5)(6,8)(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,c^a=cf,c^b=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11409, #479915, #479916, … |
| maximal normal subgroup of | C2×#11409, #479915, #479916, … |
| maximal quotient of | #126527, #126561, #128210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4397);