Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C23 | C23×C4 |
| Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1287 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 1287 conjugacy classes of subgroups — too many to draw. Only its 311 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:
(1,2)(3,4)(9,10,11,12)(13,14)(15,18)(16,17)(19,20);(2,4)(5,6)(7,8)(14,17)(18,20);(2,4)(6,8);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|a^2=h,d^2=ef,b^a=be,c^a=cf,c^b=cg88 conjugacy classes — view the full character table.
| maximal subgroup of | #106832, #107133, #110602, … |
| maximal normal subgroup of | #106832, #107133, #110602, … |
| maximal quotient of | #153734, #153739, C4×#1438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6835);