Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
723 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 723 conjugacy classes of subgroups — too many to draw. Only its 231 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) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(6,9)(10,12)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(15,21)(18,24)(22,27)(25,28);(14,19)(18,24)(20,26)(25,28)Polycyclic presentation:
2^8|a^2=g,c^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #101230, #101234, #101238, … |
| maximal normal subgroup of | #101230, #101234, #101238, … |
| maximal quotient of | #189721, #189722, #189723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13671);