Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3519 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
523 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(4,12)(7,14)(10,16)(13,15)(18,23)(22,30)(24,31)(25,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)Polycyclic presentation:
2^8|a^2=f,d^2=g,e^2=h,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dg49 conjugacy classes — view the full character table.
| maximal subgroup of | #221364, #221365, #221370, … |
| maximal normal subgroup of | #221364, #221365, #221370, … |
| maximal quotient of | #160632, #160639, #160645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10587);