Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1007 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 183 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 — 4 generators on 20 points:
(5,6)(7,15)(8,12)(9,13)(10,14)(11,16)(17,19)(18,20);(1,2)(3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,15)(13,19)(14,20);(2,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,20,15)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,d^b=dh,e^b=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #394020, #394021, #394113, … |
| maximal normal subgroup of | #394020, #394021, #394113, … |
| maximal quotient of | #232219, #232220, #232225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16397);