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 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,29)(22,25)(23,28)(27,32)(30,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,27)(23,31)(26,32)(28,30);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,23)(20,28)(21,30)(22,27)(24,32)(25,31)(26,29);(2,8)(6,9)(7,16)(14,15);(17,20,22,21)(18,24,25,29)(19,28,27,30)(23,32,31,26)Polycyclic presentation:
2^8|b^2=f,c^2=f,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 | #113136, #113165, #113236, … |
| maximal normal subgroup of | #113136, #113165, #113236, … |
| maximal quotient of | #6374364, #6374442, #6374971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53842);