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) = C27 |
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 | |
1082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,21)(22,23);(2,5)(6,7)(9,11,14,12)(10,15,13,16)(17,19,22,20)(18,21,23,24);(11,12)(15,16)(18,23)(19,20);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,13,16)(17,20,22,19)(18,21,23,24);(1,3,4,8)(2,6,5,7)(9,13,14,10)(11,15,12,16)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=fg,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×#51113, #7707097, #7707159, … |
| maximal normal subgroup of | C2×#51113, #7707097, #7707159, … |
| maximal quotient of | #560871, #561011, #561980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44101);