Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = C25 · Out(G) = C24×D4 |
| 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 | #537 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
832 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,25,22,26,20,30)(19,27,28,32,29,24,31,21);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,29)(18,24,26,27)(20,31,23,28)(21,30,32,25);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,22,23)(18,25,26,30)(19,28,29,31)(21,24,32,27);(2,8)(6,9)(7,16)(14,15)(18,26)(19,29)(25,30)(28,31);(17,21,23,27,22,32,20,24)(18,19,25,28,26,29,30,31)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cf,d^b=dh,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6345423, #6345424, #6345429, … |
| maximal normal subgroup of | #6345423, #6345424, #6345429, … |
| maximal quotient of | #6370514, #6370517, #6370583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53332);