Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,22,24)(19,25,27,32)(20,29,28,23)(21,31,30,26);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,28,22,27,30,20)(18,23,26,25,24,29,31,32);(2,6)(5,7)(11,15)(13,16)(18,24)(19,27)(20,28)(26,31);(2,6)(5,7)(17,20,30,27,22,28,21,19)(18,25,31,23,24,32,26,29)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=ef,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28070, #6457392, #6457411, … |
| maximal normal subgroup of | C2×#28070, #6457392, #6457411, … |
| maximal quotient of | #361190, #362401, #367736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21058);