Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#472 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,22)(9,16)(10,13)(11,15)(12,19)(14,23)(18,32)(20,28)(21,26)(24,30)(25,29)(27,34)(31,33)(35,36);(5,7,9,19,10,20,11,21)(6,12,15,17,13,26,16,28)(8,18,23,33,24,34,25,35)(14,27,22,31,29,32,30,36);(1,2,3,4)(6,13)(12,26)(14,29)(15,16)(17,28)(22,30)(27,32)(31,36);(5,8)(6,14)(7,18)(9,23)(10,24)(11,25)(12,27)(13,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(26,32)(28,36)Polycyclic presentation:
2^8|b^2=e,c^2=fh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138144, #138158, #138179, … |
| maximal normal subgroup of | #138144, #138158, #138179, … |
| maximal quotient of | #351644, #351645, #351652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26180);