Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(3,9)(4,11)(7,14)(8,12)(13,15)(17,18,21,25,22,24,20,29)(19,26,27,23,28,32,30,31);(1,2,5,8)(3,7,9,15)(4,12,11,6)(10,14,16,13)(17,19,22,28)(18,23,24,31)(20,30,21,27)(25,32,29,26);(1,3)(2,7)(4,10)(5,9)(6,13)(8,15)(11,16)(12,14)(18,24)(23,31)(25,29)(26,32);(3,9)(7,15)(10,16)(13,14)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,e^a=eg,e^b=eg,d^c=dgh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17524, #6343069, #6343077, … |
| maximal normal subgroup of | C2×#17524, #6343069, #6343077, … |
| maximal quotient of | #164404, #164405, #164408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10512);