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) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#121 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 599 conjugacy classes of subgroups — too many to draw. Only its 207 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(1,2,3,4)(13,15,16,22,17,21,23,27)(14,18,19,25,20,24,26,28);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=f,g^2=h,b^a=be,c^a=cf,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #110584, #110585, #110586, … |
| maximal normal subgroup of | #110584, #110585, #110586, … |
| maximal quotient of | #207480, #207481, #207482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14365);