Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(11,15)(14,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32);(9,10)(11,14)(12,13)(15,16)(18,19)(20,27)(24,26)(28,32);(2,3)(6,7)(10,13)(14,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52470, #8092148, #8092171, … |
| maximal normal subgroup of | C2×#52470, #8092148, #8092171, … |
| maximal quotient of | #582287, #582492, #582532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45458);