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 | |
380 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(2,4)(5,6,8,9)(7,11,12,10)(13,14,16,19)(15,21,23,18)(17,20,25,28)(22,27,24,26);(6,9)(10,11)(13,15,17,24,16,23,25,22)(14,18,20,27,19,21,28,26);(1,2,3,4)(15,22)(17,25)(18,26)(20,28)(21,27)(23,24);(5,7,8,12)(6,10,9,11)(14,19)(18,21)(20,28)(26,27)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=f,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 | #263755, #263756, #263797, … |
| maximal normal subgroup of | #263755, #263756, #263797, … |
| maximal quotient of | #324785, #324786, #324787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21629);