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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32);(1,2)(3,7)(4,8)(5,6)(9,11,13,10)(12,15,16,14)(18,19)(21,29)(23,25)(27,32);(2,6)(3,8)(9,12)(10,14)(11,15)(13,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(1,3)(2,4)(5,8)(6,7)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,4)(2,3)(5,7)(6,8)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=f,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×#47902, #7703347, #7704095, … |
| maximal normal subgroup of | C2×#47902, #7703347, #7704095, … |
| maximal quotient of | #560415, #560565, #561423, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40890);