Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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×C4⋊C4 |
| prime radical | C2×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,15)(6,9)(7,13)(11,12)(14,16)(17,18,21,25,22,26,20,24)(19,27,30,31,28,32,29,23);(1,3,4,12,6,14,5,13)(2,7,15,16,9,11,8,10)(17,19,21,30,22,28,20,29)(18,23,24,32,26,31,25,27);(1,2)(3,11)(4,15)(5,8)(6,9)(7,14)(10,12)(13,16)(19,28)(23,31)(27,32)(29,30);(1,4,6,5)(2,8,9,15)(3,12,14,13)(7,10,11,16)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,32,31,27)Polycyclic presentation:
2^8|a^2=f,b^2=eg,d^2=g,e^2=g,f^2=h,b^a=be,e^a=eg,c^b=cg,d^c=dgh58 conjugacy classes — view the full character table.
| maximal subgroup of | #225788, #225789, #225792, … |
| maximal normal subgroup of | #225788, #225789, #225792, … |
| maximal quotient of | #180940, #183424, #183594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10829);