Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| 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 |
| prime radical | #3188 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
866 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)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,28,20,24)(21,27,23,31)(25,29,32,30);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,24)(20,22)(21,30)(23,29)(25,31)(26,28)(27,32);(3,10)(6,9)(11,16)(13,15)(17,20,22,19)(18,25,26,32)(21,30,23,29)(24,27,28,31);(2,7)(6,9)(8,14)(13,15)(18,26)(24,28)(25,32)(27,31);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(17,21,22,23)(18,27,26,31)(19,29,20,30)(24,25,28,32)Polycyclic presentation:
2^8|a^2=h,c^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6352710, #6352730, #6352740, … |
| maximal normal subgroup of | #6352710, #6352730, #6352740, … |
| maximal quotient of | #6374724, #6374729, #6375095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53969);