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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
768 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)(3,5)(4,8)(6,7)(9,10,13,16)(11,14,12,15)(18,19)(21,29)(23,25)(27,32);(1,3)(2,5)(4,6)(7,8)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(2,7)(3,4)(9,10)(11,14)(12,15)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,4,6,3)(2,8,7,5)(9,11,13,12)(10,14,16,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,5,6,8)(2,3,7,4)(9,12,13,11)(10,15,16,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,c^2=f,d^2=fg,e^2=fg,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×#49399, #7704953, #7704977, … |
| maximal normal subgroup of | C2×#49399, #7704953, #7704977, … |
| maximal quotient of | #560690, #560912, #561773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42387);