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)(12,15)(16,20)(17,21)(19,22)(23,24);(1,3,4,8)(2,5,6,7);(9,10,15,21,13,17,12,14)(11,16,22,24,18,23,19,20);(9,11)(10,16)(12,19)(13,18)(14,20)(15,22)(17,23)(21,24);(11,18)(16,23)(19,22)(20,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=gh,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 | #60667⋊C3 |
| maximal subgroup of | #130015, #130016, #250025, … |
| maximal normal subgroup of | #130015, #130016, #250025, … |
| maximal quotient of | #6371140, #6371876, #6372211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53655);