Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C2×#120 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3535 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,9,11,16)(4,12,14,8)(6,13,15,10)(17,18)(19,22)(20,23)(21,24)(25,31)(26,29)(27,30)(28,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(2,7)(6,15)(8,12)(9,16)(18,24)(22,29)(23,30)(28,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18449, #6347063, #6347083, … |
| maximal normal subgroup of | C2×#18449, #6347063, #6347083, … |
| maximal quotient of | #168912, #168924, #168967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11437);