Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C24 | C22×C4 |
| Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(1,2)(3,4)(13,14)(15,19,20,17)(16,18);(5,6,8,10)(7,11,12,9)(13,15)(14,17)(16,20)(18,19);(2,4)(5,7,8,12)(6,9,10,11)(13,14)(15,17)(16,18)(19,20)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=eh,g^b=gh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #11390⋊C3, #72227677 |
| maximal subgroup of | #93976, #137307, #137308, … |
| maximal normal subgroup of | #93976, #137307, #137308, … |
| maximal quotient of | #121985, #122003, #122005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4378);