Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| 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 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
955 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 955 conjugacy classes of subgroups — too many to draw. Only its 487 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:
(2,5)(3,7)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19);(1,2)(3,6)(4,5)(7,8);(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=h,c^2=gh,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df64 conjugacy classes — view the full character table.
| Fitting subgroup of | #62416⋊C3 |
| maximal subgroup of | #131652, #131656, #250112, … |
| maximal normal subgroup of | #131652, #131656, #250112, … |
| maximal quotient of | #6386874, #6389485, #6389848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55404);