Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| 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×#550 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups; only the 481 normal subgroups are shown.
This group has 1019 conjugacy classes of subgroups — too many to draw. Only its 481 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) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24);(2,5)(3,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24);(10,18)(15,19)(16,17)(23,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,g^c=gh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #61881⋊C3 |
| maximal subgroup of | #245724, #245727, #245748, … |
| maximal normal subgroup of | #245724, #245727, #245748, … |
| maximal quotient of | #6386307, #6388179, #6392757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54869);