Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | |
850 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)(3,6)(4,7)(5,8)(10,13)(14,16)(18,22)(23,24);(1,2)(3,6)(4,7)(5,8)(17,18)(19,23)(20,24)(21,22);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,23,22,24);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23);(1,4,5,6)(2,7,8,3)Polycyclic presentation:
2^8|c^2=f,d^2=fgh,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6580243, #6580246, #6580306, … |
| maximal normal subgroup of | #6580243, #6580246, #6580306, … |
| maximal quotient of | #517778, #517784, #517798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28714);