Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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 | |
370 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 123 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)(15,18)(16,20)(17,19)(21,25)(22,26)(23,28)(24,27);(6,9)(10,11)(13,15,21,22,17,24,16,23)(14,18,25,26,19,27,20,28);(1,2,3,4)(15,22)(16,21)(18,26)(20,25)(23,24)(27,28);(5,7,8,12)(6,10,9,11)(14,19)(18,27)(20,25)(26,28)Polycyclic presentation:
2^8|a^2=e,b^2=gh,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #263775, #263776, #263817, … |
| maximal normal subgroup of | #263775, #263776, #263817, … |
| maximal quotient of | #324883, #324884, #324885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21632);