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 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(11,15)(14,16)(18,22)(23,24);(2,3)(6,7)(10,13)(11,15);(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,20,24);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,19,21,20)(18,23,22,24);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=fgh,e^2=g,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×#46124, #7702576, #7702717, … |
| maximal normal subgroup of | C2×#46124, #7702576, #7702717, … |
| maximal quotient of | #557168, #557232, #557486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39112);