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 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
965 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 965 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) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,21,23,18)(16,24,25,19)(22,26,28,27);(5,6,8,10)(7,9,12,11)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27);(1,2,3,4)(5,6)(7,9)(8,10)(11,12);(5,7)(6,9)(8,12)(10,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28);(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)Polycyclic presentation:
2^8|a^2=fh,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 | #97279, #131388, #167777, … |
| maximal normal subgroup of | #97279, #131388, #167777, … |
| maximal quotient of | #553756, #554015, #554325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30375);