Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C23×C8 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
489 conjugacy classes of subgroups; only the 279 normal subgroups are shown.
This group has 489 conjugacy classes of subgroups — too many to draw. Only its 279 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,15,17,16,18,19,20);(5,7,8,12)(6,9,10,11);(1,2,3,4)(7,12)(9,11);(6,10)(9,11)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #99316, #99317, #99549, … |
| maximal normal subgroup of | #99316, #99317, #99549, … |
| maximal quotient of | #217364, #217365, #217366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14939);