Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
989 conjugacy classes of subgroups; only the 463 normal subgroups are shown.
This group has 989 conjugacy classes of subgroups — too many to draw. Only its 463 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:
(1,2,3,4)(7,11)(10,12)(14,17)(16,24)(19,22)(21,25);(5,6)(7,10)(8,9)(11,12)(13,14,18,17)(15,19,23,22)(16,20,24,26)(21,27,25,28);(1,2)(3,4)(6,9)(10,12)(13,15,18,23)(14,19,17,22)(16,25,24,21)(20,28,26,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,21)(17,26)(18,24)(19,27)(22,28)(23,25);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,22,23,19)(16,26,24,20)(21,28,25,27)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #111235, #111260, #111998, … |
| maximal normal subgroup of | #111235, #111260, #111998, … |
| maximal quotient of | #556345, #556383, #556567, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30629);