Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | Aut(G) = #72305173 (order 4096) · Inn(G) = C2×#102 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
255 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,10,12,14)(7,11)(8,16)(13,18,20,17)(15,19);(5,6,11,18,16,20,7,14,9,12,19,17,8,13,15,10);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130210, #130211, #130244, … |
| maximal normal subgroup of | #130210, #130211, #130244, … |
| maximal quotient of | #137611, #137612, #137613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6053);