Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C23×#6342371 (order 4096) · Inn(G) = C25 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #537 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
833 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,20,24,21,22,27,29)(19,23,26,30,25,28,31,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,25)(23,28)(26,31)(30,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,22)(23,28)(24,29)(30,32);(2,8)(6,9)(7,16)(14,15);(17,19,20,26,21,25,27,31)(18,23,24,30,22,28,29,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cf,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129675, #209594, #209595, … |
| maximal normal subgroup of | #129675, #209594, #209595, … |
| maximal quotient of | #6370581, #6370586, #6370596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53328);