Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| 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 | C24 | C24 |
| socle | C22 | C2×#120 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1129 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 1129 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:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,16)(15,18)(19,20);(10,16)(17,20);(2,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20);(5,8,9,15)(6,11,12,18)(7,13,14,19)(10,17,16,20)Polycyclic presentation:
2^8|d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137133, #137134, #137204, … |
| maximal normal subgroup of | #137133, #137134, #137204, … |
| maximal quotient of | #232207, #232208, #232209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16431);