Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
930 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)(17,18,22,21)(19,24,23,20);(1,3,6,4)(2,7,8,5)(9,10,12,14)(11,13,16,15)(17,19)(18,20)(21,24)(22,23);(2,8)(3,4)(9,10)(11,15)(12,14)(13,16)(17,20,22,24)(18,23,21,19);(1,4,6,3)(2,5,8,7)(9,11)(10,13)(12,16)(14,15);(1,5)(2,4)(3,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,21,22,18)(19,24,23,20)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#36337, #7629993, #7630014, … |
| maximal normal subgroup of | C2×#36337, #7629993, #7630014, … |
| maximal quotient of | #519201, #519203, #519220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29325);