Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
661 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 661 conjugacy classes of subgroups — too many to draw. Only its 159 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:
(5,6,8,9)(7,10,11,12)(13,14,15,18,16,19,20,17);(1,2)(3,4)(6,9)(10,12)(14,17)(15,20)(18,19);(2,4)(14,17)(15,20)(18,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=e,d^2=efh,e^2=h,f^2=h,b^a=be,c^a=cf,c^b=cg,e^b=eh,f^b=fh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #99675, #102541, #102543, … |
| maximal normal subgroup of | #99675, #102541, #102543, … |
| maximal quotient of | #244117, #244141, #244165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16786);