Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22284 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2328 · Out(G) = C24 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2328 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3154 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,17,21,27)(6,12)(7,13)(8,15)(9,20,28,31)(10,22)(11,24)(14,29,30,25)(16,23)(18,19,26,32);(1,3,6,15,7,16,24,10)(2,8,11,5,13,22,12,23)(4,18,31,14,21,26,20,30)(9,19,17,25,28,32,27,29);(1,4,7,21)(2,9)(3,14)(5,19,23,32)(6,20,24,31)(8,25,22,29)(10,26)(11,27)(12,17)(13,28)(15,18)(16,30);(1,5,6,22,7,23,24,8)(2,10,11,3,13,15,12,16)(4,19,20,29,21,32,31,25)(9,26,27,14,28,18,17,30)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=ef,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22284, #6376468, #6376469, … |
| maximal normal subgroup of | C2×#22284, #6376468, #6376469, … |
| maximal quotient of | #219484, #219485, #219489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15272);