Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2)(3,4)(6,10)(7,13)(8,15)(11,19)(12,20)(14,17)(16,18);(5,6)(7,14)(8,17)(9,11)(10,13)(12,15)(16,20)(18,19);(2,4)(6,11)(10,19)(12,17)(14,20);(5,7)(6,12)(8,18)(9,16)(10,19)(11,17)(13,15)Polycyclic presentation:
2^8|e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gh,c^b=ch,d^b=de,e^b=eg,g^b=gh,e^d=eg,g^d=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #491728, #491732, #491737, … |
| maximal normal subgroup of | #491728, #491732, #491737, … |
| maximal quotient of | #491764, #491769, #491828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26826);