Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
518 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)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,27,29,32)(20,25)(21,26)(22,24)(23,28,31,30);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,24)(23,31)(25,26)(27,32);(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=h,e^2=h,b^a=be,c^a=cgh,e^a=eg,e^b=eh,f^b=fgh64 conjugacy classes — view the full character table.
| maximal subgroup of | #106212, #107326, #107333, … |
| maximal normal subgroup of | #106212, #107326, #107333, … |
| maximal quotient of | #160453, #160471, #160472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10561);