Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,5)(3,8,4,7)(18,21)(22,24);(2,5)(7,8)(10,14)(12,15)(18,21)(19,23);(1,3,6,4)(2,7,5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(1,4,6,3)(2,8,5,7)(9,11,13,16)(10,15,14,12);(1,5,6,2)(3,8,4,7)(9,12,13,15)(10,16,14,11)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#40361, #7637584, #7638137, … |
| maximal normal subgroup of | C2×#40361, #7637584, #7638137, … |
| maximal quotient of | #522019, #522541, #523093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33349);