Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | |
809 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)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,21,20)(19,23,24,22);(2,6)(3,8)(10,13)(11,16);(9,10,12,13)(11,14,16,15)(17,19,21,24)(18,22,20,23);(1,3,5,8)(2,7,6,4)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,23,24,22);(1,4,5,7)(2,8,6,3)(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^8|a^2=gh,c^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#43736, #7667899, #7668072, … |
| maximal normal subgroup of | C2×#43736, #7667899, #7668072, … |
| maximal quotient of | #535530, #535752, #536253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 36724);