Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 426 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)(15,21)(16,24)(19,22)(20,26)(23,27);(5,7,8,11)(6,9,10,12)(13,14,16,18,17,20,24,26)(15,22,21,28,23,19,27,25);(6,10)(9,12)(15,23)(19,22)(21,27)(25,28);(2,4)(13,15,16,21,17,23,24,27)(14,19,18,25,20,22,26,28)Polycyclic presentation:
2^8|b^2=ef,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 | #265345, #265346, #265377, … |
| maximal normal subgroup of | #265345, #265346, #265377, … |
| maximal quotient of | #351360, #351361, #351366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26125);