Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
769 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)(3,7,8,4)(10,12)(13,15)(17,18,20,22)(19,23,24,21);(3,8)(4,7)(9,10,14,12)(11,13,16,15);(1,3)(2,4)(5,8)(6,7)(9,11,14,16)(10,15,12,13)(17,19,20,24)(18,21,22,23);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,13,16,15);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,16)(14,15)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=gh,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97874, #172335, #172336, … |
| maximal normal subgroup of | #97874, #172335, #172336, … |
| maximal quotient of | #557099, #557104, #557397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38782);