Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3536 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
596 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,8)(9,10,15,20)(11,21,12,17)(13,19,14,18)(16,23,24,22);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,20,24)(13,22,14,23)(17,18,21,19);(1,3)(2,5)(4,7)(6,8)(9,12,15,11)(10,17,20,21)(13,22,14,23)(16,18,24,19);(9,13,15,14)(10,18,20,19)(11,22,12,23)(16,21,24,17)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #249880, #249907, #249914, … |
| maximal normal subgroup of | #249880, #249907, #249914, … |
| maximal quotient of | #167000, #167018, #167063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11249);