Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | C2×H(5,𝔽2) | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,15,12,11,14,16);(1,3)(2,5)(4,8)(6,7)(10,11)(15,16);(3,8)(5,7)(10,11)(15,16);(2,6)(5,7);(9,11,13,16,12,10,14,15)Polycyclic presentation:
2^8|a^2=g,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=dfh,c^b=cfh,e^b=eh,e^c=eh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #60325⋊C3 |
| maximal subgroup of | #129659, #209384, #209385, … |
| maximal normal subgroup of | #129659, #209384, #209385, … |
| maximal quotient of | #6370409, #6370418, #6370509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53313);