Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
829 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 829 conjugacy classes of subgroups — too many to draw. Only its 343 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19);(5,7,9,17,10,16,8,15)(6,11,12,19,13,14,18,20);(1,3,4,2)(5,6)(7,11)(8,18)(9,12)(10,13)(14,16)(15,20)(17,19);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,g^2=h,b^a=be,g^a=gh,e^b=eh,d^c=dh82 conjugacy classes — view the full character table.
| maximal subgroup of | #138243, #138310, #138312, … |
| maximal normal subgroup of | #138243, #138310, #138312, … |
| maximal quotient of | #181945, #181947, #181958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13386);