Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| 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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,21)(18,23,26,24)(20,28,29,27)(25,31,32,30);(3,10)(6,9)(11,16)(13,15)(18,24)(19,21)(23,26)(25,31)(27,28)(30,32);(2,7)(6,9)(8,14)(13,15)(17,20,22,29)(18,25,26,32)(19,27,21,28)(23,30,24,31);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(20,29)(25,32)(27,28)(30,31)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,d^b=dh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6352021, #6352022, #6352023, … |
| maximal normal subgroup of | #6352021, #6352022, #6352023, … |
| maximal quotient of | #6374697, #6374700, #6375047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54009);