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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,21)(22,23)(25,30)(26,31)(27,32)(28,29);(1,2,4,6)(3,5,8,7)(18,23)(19,26)(21,30)(27,29);(9,10)(11,13)(12,14)(15,16)(17,19,22,26)(18,24,23,31)(20,29,25,27)(21,28,30,32);(1,2)(3,5)(4,6)(7,8)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=h,d^2=h,e^2=g,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×#51490, #7707320, #7707346, … |
| maximal normal subgroup of | C2×#51490, #7707320, #7707346, … |
| maximal quotient of | #560779, #561074, #561899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44478);