Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
493 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,10,8,12,7,11)(13,14,15,18,16,17,20,19);(1,2,3,4)(14,17)(18,19);(2,4)(6,10)(7,9)(11,12)(13,14)(15,19)(16,17)(18,20);(13,15,16,20)(14,18,17,19)Polycyclic presentation:
2^8|a^2=efh,b^2=eg,d^2=e,f^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #103957, #103961, #103965, … |
| maximal normal subgroup of | #103957, #103961, #103965, … |
| maximal quotient of | #331514, #331520, #331527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23265);