Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,16)(11,13,12,15)(17,18,22,23)(19,24,21,20);(1,2,4,5)(3,7,8,6)(9,11,14,12)(10,15,16,13)(18,23)(19,21);(2,5)(6,7)(10,16)(11,12)(17,19,22,21)(18,24,23,20);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,16,13)(17,20,22,24)(18,19,23,21);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,11,16,12)(17,21,22,19)(18,24,23,20)Polycyclic presentation:
2^8|a^2=f,b^2=fgh,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#51917, #7707952, #7707955, … |
| maximal normal subgroup of | C2×#51917, #7707952, #7707955, … |
| maximal quotient of | #560343, #560511, #561538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44905);