Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 139 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)(5,6,9,12)(7,11,8,10)(13,14,16,19)(15,21,24,18)(17,20,22,26)(23,28,25,27);(5,7,9,8)(6,10,12,11)(13,15,22,23,16,24,17,25)(14,18,26,27,19,21,20,28);(2,4)(15,23)(17,22)(18,27)(20,26)(21,28)(24,25);(5,8,9,7)(6,11,12,10)(14,19)(18,21)(20,26)(27,28)Polycyclic presentation:
2^8|a^2=e,b^2=eg,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #264631, #264634, #264637, … |
| maximal normal subgroup of | #264631, #264634, #264637, … |
| maximal quotient of | #318683, #318684, #318685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21254);