Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
447 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,22)(16,19)(20,23)(21,26)(24,27);(5,7,8,11)(6,9,10,12)(13,14,19,26,17,21,16,18)(15,23,27,25,24,20,22,28);(6,10)(9,12)(15,24)(20,23)(22,27)(25,28);(2,4)(13,15,16,22,17,24,19,27)(14,20,18,25,21,23,26,28)Polycyclic presentation:
2^8|b^2=efh,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #144256, #144258, #265347, … |
| maximal normal subgroup of | #144256, #144258, #265347, … |
| maximal quotient of | #351362, #351363, #351370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26126);