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 | #2414 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,4,8,7)(9,10,13,15)(11,12,16,14)(17,18,21,24)(19,22,20,23);(2,6)(4,7)(9,11,13,16)(10,14,15,12)(17,19,21,20)(18,22,24,23);(2,6)(3,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,14)(15,16)(17,20,21,19)(18,23,24,22);(1,4,5,7)(2,8,6,3)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,d^2=g,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 | #6696349, #6696352, #6696412, … |
| maximal normal subgroup of | #6696349, #6696352, #6696412, … |
| maximal quotient of | #544602, #545143, #545630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35728);