Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1011 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 1011 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,15)(8,17)(9,12)(10,13)(11,14)(16,19)(18,20);(1,2)(3,4)(5,7)(6,8)(9,16)(10,17)(11,18)(12,15)(13,20)(14,19);(2,4)(5,8)(6,7)(9,15)(10,20)(11,19)(12,16)(13,17)(14,18);(5,9,11,10)(6,12,14,13)(7,16,18,17)(8,15,19,20)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,e^b=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137139, #137140, #137209, … |
| maximal normal subgroup of | #137139, #137140, #137209, … |
| maximal quotient of | #232199, #232206, #232209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16394);