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 | #2428 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
992 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)(11,13)(12,15)(14,16);(1,2,5,3)(4,7,8,6)(9,11,14,12)(10,15,16,13)(17,18)(19,20)(21,22)(23,24);(2,3)(6,7)(10,16)(11,12)(18,22)(19,23);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,15,16,13)(17,19,21,23)(18,20,22,24);(1,4)(2,6)(3,7)(5,8)(9,13,14,15)(10,11,16,12)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^8|a^2=fg,b^2=g,d^2=h,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 | #6455289, #6455295, #6455307, … |
| maximal normal subgroup of | #6455289, #6455295, #6455307, … |
| maximal quotient of | #515385, #515409, #515439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28548);