Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,8,7,5)(9,10,14,13)(11,16,15,12)(17,18,21,20)(19,23,24,22);(2,4)(3,7)(9,11)(10,12)(13,16)(14,15)(17,19,21,24)(18,22,20,23);(1,3)(2,5)(4,8)(6,7)(9,12,14,16)(10,15,13,11)(17,18,21,20)(19,22,24,23);(1,4,6,2)(3,8,7,5)(9,10,14,13)(11,12,15,16);(1,5,6,8)(2,7,4,3)(9,13,14,10)(11,16,15,12)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #161608, #161892, #162388, … |
| maximal normal subgroup of | #161608, #161892, #162388, … |
| maximal quotient of | #516512, #516540, #516690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28105);