Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | Aut(G) order 24576 |
| 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 | C42 | C24 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(18,23,25,29)(24,30,32,31);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,32)(28,30);(17,19,22,27)(18,24,25,32)(20,28,21,26)(23,30,29,31);(17,20,22,21)(18,23,25,29)(19,26,27,28)(24,31,32,30)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=h,e^2=h,f^2=g,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #17692⋊C3 |
| maximal subgroup of | #220606, #220607, #220609, … |
| maximal normal subgroup of | #220606, #220607, #220609, … |
| maximal quotient of | #163742, #163778, #163821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10680);