Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1309 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 1309 conjugacy classes of subgroups — too many to draw. Only its 455 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 — 5 generators on 20 points:
(2,4)(6,9)(7,11)(15,19)(18,20);(5,6,8,9)(7,10,11,12)(14,17)(18,20);(1,2,3,4)(13,14)(15,18)(16,17)(19,20);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|b^2=f,c^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97271, #131387, #167775, … |
| maximal normal subgroup of | #97271, #131387, #167775, … |
| maximal quotient of | #553755, #554014, #554324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30286);