Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,13)(14,16)(17,18,23,21)(19,26,20,24)(22,29,30,25)(27,31,28,32);(2,5)(3,7)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,21,23,18)(19,24,20,26)(22,29,30,25)(27,31,28,32);(1,2)(3,6)(4,5)(7,8)(17,22,23,30)(18,25,21,29)(19,27,20,28)(24,31,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45559, #7702252, #7702258, … |
| maximal normal subgroup of | C2×#45559, #7702252, #7702258, … |
| maximal quotient of | #557128, #557208, #557420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38547);