Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 139 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)(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,24)(17,26)(18,21)(20,25)(22,28)(23,27);(6,9)(10,11)(13,15,18,23)(14,19,21,27)(16,25,22,17)(20,28,26,24);(1,2,3,4)(13,16,17,15,18,22,25,23)(14,19,20,24,21,27,26,28);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138779, #250910, #250990, … |
| maximal normal subgroup of | #138779, #250910, #250990, … |
| maximal quotient of | #295845, #295856, #295863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18689);