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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
794 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)(4,7,8,6)(9,10,13,14)(11,15,16,12)(19,23)(22,24);(2,3)(6,7)(10,14)(11,16)(17,18,21,20)(19,22,23,24);(9,11,13,16)(10,15,14,12)(17,19)(18,22)(20,24)(21,23);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,16,14,11)(17,18,21,20)(19,22,23,24);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=fg,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×#45929, #7702367, #7702514, … |
| maximal normal subgroup of | C2×#45929, #7702367, #7702514, … |
| maximal quotient of | #557312, #557324, #557517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38917);