Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #2416 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,16)(12,14)(13,15)(19,23)(21,24);(1,2,5,3)(4,7,8,6)(9,10,14,12)(11,15,13,16)(17,18,20,22)(19,21,23,24);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(17,19)(18,21)(20,23)(22,24);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,12,16)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=f,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6678383, #6678386, #6678446, … |
| maximal normal subgroup of | #6678383, #6678386, #6678446, … |
| maximal quotient of | #544427, #545015, #545495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35934);