Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(11,17)(12,20)(14,21)(16,22)(18,23)(19,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,18)(12,16)(13,15)(14,19)(17,23)(20,22)(21,24);(9,11,12,17,13,19,20,24)(10,14,16,21,15,18,22,23);(3,8)(5,7);(2,6)(5,7)(10,15)(14,18)(16,22)(21,23)Polycyclic presentation:
2^8|c^2=g,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62277, #10577582, #10577658, … |
| maximal normal subgroup of | C2×#62277, #10577582, #10577658, … |
| maximal quotient of | #6386752, #6389140, #6389144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55265);