Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| 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 | |
814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,6,8,7)(10,13)(14,16)(17,18,20,22)(19,23,24,21);(1,3)(2,6)(4,7)(5,8)(17,19,20,24)(18,21,22,23);(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23);(9,11,12,15)(10,14,13,16)(17,18)(19,21)(20,22)(23,24);(1,4,5,2)(3,6,8,7)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fh,d^2=fg,e^2=fh,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 | #6700404, #6700409, #6700469, … |
| maximal normal subgroup of | #6700404, #6700409, #6700469, … |
| maximal quotient of | #521447, #522211, #522620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32526);