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 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
812 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)(3,8)(4,5)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18,21,20)(19,23,24,22);(1,3,6,4)(2,5,7,8)(11,13)(12,15);(2,7)(3,4)(9,11,14,13)(10,15,16,12)(17,19,21,24)(18,22,20,23);(1,4,6,3)(2,5,7,8)(9,12,14,15)(10,11,16,13)(17,18,21,20)(19,22,24,23);(1,5)(2,3)(4,7)(6,8)(9,13,14,11)(10,15,16,12)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=fg,d^2=g,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×#49490, #7704719, #7704979, … |
| maximal normal subgroup of | C2×#49490, #7704719, #7704979, … |
| maximal quotient of | #560862, #560988, #561969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42478);