Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,12)(8,10)(9,11)(13,14,17,19,18,15,16,20);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,14)(16,19,17,20);(2,4)(5,8,9,7)(6,10,11,12)(13,14,18,15)(16,20,17,19)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,e^b=eh,d^c=dh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #92890, #93912, #97321, … |
| maximal normal subgroup of | #92890, #93912, #97321, … |
| maximal quotient of | #121616, #121617, #121630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4347);