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 | |
837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10)(11,12)(13,15)(14,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30);(1,2)(3,7)(4,8)(5,6)(11,13)(12,15)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30);(2,6)(3,8)(9,11,14,13)(10,15,16,12)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30);(1,3)(2,4)(5,8)(6,7)(9,12,14,15)(10,11,16,13)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,15,16,12)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,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×#48678, #7703789, #7704116, … |
| maximal normal subgroup of | C2×#48678, #7703789, #7704116, … |
| maximal quotient of | #560797, #561113, #561917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41666);