Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,15,19,16,17,20);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15,17,14,19,20,18,16);(6,10)(9,12)(13,16,19,14)(15,17,20,18);(1,3,4,2)Polycyclic presentation:
2^8|a^2=g,b^2=ef,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #311033, #311041, #311049, … |
| maximal normal subgroup of | #311033, #311041, #311049, … |
| maximal quotient of | #315025, #315026, #315027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20757);