Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
587 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(13,14,17,21,18,22,16,20)(15,19,23,26,24,27,25,28);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,25)(17,23)(18,24)(20,28)(21,26)(22,27);(2,4)(13,16,18,17)(14,20,22,21)(15,23,24,25)(19,26,27,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,f^b=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #93107, #93108, #93112, … |
| maximal normal subgroup of | #93107, #93108, #93112, … |
| maximal quotient of | #96932, #96937, #96976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1296);