Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#393 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
574 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)(3,8)(4,16,18,27)(5,10)(6,21)(7,11)(9,19,26,20)(12,25,28,32)(13,23)(14,24)(15,30)(17,22,29,31);(1,3)(2,8)(4,17,18,29)(5,14)(6,15)(7,13)(9,25,26,32)(10,24)(11,23)(12,20,28,19)(16,31,27,22)(21,30);(1,4)(2,9)(3,12)(5,20)(6,19)(7,18)(8,22)(10,27)(11,26)(13,28)(14,17)(15,29)(16,21)(23,31)(24,32)(25,30);(3,13)(8,23)(12,28)(14,15)(17,29)(22,31)(24,30)(25,32)Polycyclic presentation:
2^8|a^2=e,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=dh,e^c=eh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #394482, #394483, #394484, … |
| maximal normal subgroup of | #394482, #394483, #394484, … |
| maximal quotient of | #237956, #237957, #237958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16460);