Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,20)(19,24)(22,23);(1,3)(2,5)(4,8)(6,7)(11,19)(15,23)(18,24)(20,22);(3,8)(5,7)(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21);(2,6)(5,7);(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,24,23,18)Polycyclic presentation:
2^8|e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #111405, #111455, #111482, … |
| maximal normal subgroup of | #111405, #111455, #111482, … |
| maximal quotient of | #6374159, #6374868, #6375498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53729);