Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
562 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 562 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6,8,7)(9,10,13,16,12,15,11,14);(6,7)(9,10,11,14,12,15,13,16);(1,2,3,4);(5,7,8,6)(9,11,12,13)(10,14,15,16)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^b=deh76 conjugacy classes — view the full character table.
| maximal subgroup of | #99347, #99348, #99351, … |
| maximal normal subgroup of | #99347, #99348, #99351, … |
| maximal quotient of | #231452, #231453, #231454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16164);