Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
911 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)(3,5)(4,7)(6,8)(10,12)(11,13)(17,18)(19,20)(21,22)(23,24);(1,3,4,8)(2,5,7,6)(9,10,14,12)(11,16,13,15)(18,21)(19,23);(1,3)(2,6)(4,8)(5,7)(9,11,14,13)(10,15,12,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,16,13,15)(17,20,22,24)(18,23,21,19);(1,2)(3,6)(4,7)(5,8)(9,13,14,11)(10,16,12,15)(17,21,22,18)(19,24,23,20)Polycyclic presentation:
2^8|b^2=g,c^2=fgh,d^2=f,e^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 | C2×#45814, #7702501, #7702506, … |
| maximal normal subgroup of | C2×#45814, #7702501, #7702506, … |
| maximal quotient of | #557070, #557101, #557351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38802);