Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | |
932 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)(3,7,8,6)(17,18,22,21)(19,24,20,23);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,20)(18,23,21,24);(3,8)(6,7)(10,14)(11,15)(17,20,22,19)(18,24,21,23);(1,3)(2,7)(4,6)(5,8)(9,11,13,15)(10,12,14,16)(17,18,22,21)(19,23,20,24);(1,4,5,2)(3,6,8,7)(9,12)(10,15)(11,14)(13,16)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6442019, #6442021, #6442036, … |
| maximal normal subgroup of | #6442019, #6442021, #6442036, … |
| maximal quotient of | #515310, #515361, #515481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28577);