Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | #2456 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
939 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)(3,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,15)(18,23)(19,24);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,23);(1,3)(2,4)(5,7)(6,8)(9,12,13,14)(10,15,16,11)(17,19,22,24)(18,20,23,21);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,15)(14,16)(17,20,22,21)(18,19,23,24);(1,4,5,6)(2,7,8,3)(9,10)(11,14)(12,15)(13,16)(17,21,22,20)(18,24,23,19)Polycyclic presentation:
2^8|c^2=f,d^2=gh,e^2=fh,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35727, #7629334, #7629359, … |
| maximal normal subgroup of | C2×#35727, #7629334, #7629359, … |
| maximal quotient of | #517780, #517782, #517784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28715);