Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72283959 (order 1024) · Inn(G) = #99 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #99 |
| commutator | C2×C4 | C2×C16 |
| Frattini, cosocle | C2×M32 | C22 |
| socle | C2 | #1314 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,6,13,25,38,7,10,24,34,27,35,51,58,8,12,26,37,28,39,52,60,4,19,21,36,22,46,48,59)(3,14,16,40,17,41,44,61,15,9,29,30,42,32,53,55,18,33,23,49,45,57,50,63,20,31,43,54,47,56,62,64);(1,3)(2,9)(4,20)(5,23)(6,17)(7,15)(8,18)(10,33)(11,30)(12,31)(13,32)(14,19)(16,26)(21,29)(22,47)(24,43)(25,50)(27,42)(28,45)(34,49)(35,57)(36,40)(37,54)(38,55)(39,56)(41,46)(44,52)(48,53)(51,62)(58,63)(59,61)(60,64)Polycyclic presentation:
2^8|a^2=d,c^2=h,d^2=f,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eh,c^b=ch,d^b=deh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93679, #93680, #93734, … |
| maximal normal subgroup of | #93679, #93680, #93734, … |
| maximal quotient of | #93670, #93682, #93687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 325);