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, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
405 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(23,29)(26,32)(30,35)(33,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,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 | #93118, #93121, #93123, … |
| maximal normal subgroup of | #93118, #93121, #93123, … |
| maximal quotient of | #96927, #96929, #96973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1294);