Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
807 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(18,21)(22,24);(3,5)(4,7)(9,10,12,14)(11,13,16,15);(1,3,6,5)(2,7,8,4)(9,11)(10,13)(12,16)(14,15)(17,18,20,21)(19,22,23,24);(1,4,6,7)(2,3,8,5)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,22,21,24);(1,5,6,3)(2,7,8,4)(9,11)(10,13)(12,16)(14,15)Polycyclic presentation:
2^8|b^2=f,c^2=fg,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#51701, #7707431, #7707638, … |
| maximal normal subgroup of | C2×#51701, #7707431, #7707638, … |
| maximal quotient of | #561234, #561334, #562290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44689);