Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
917 conjugacy classes of subgroups; only the 563 normal subgroups are shown.
This group has 917 conjugacy classes of subgroups — too many to draw. Only its 563 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24);(1,3,4,8)(2,5,6,7);(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,22)(18,23)(20,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #60665⋊C3 |
| maximal subgroup of | #250026, #250030, #250034, … |
| maximal normal subgroup of | #250026, #250030, #250034, … |
| maximal quotient of | #6371139, #6371875, #6372211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53653);