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)(3,10,13,7)(4,14)(5,8)(6,9)(11,15,12,16)(17,18)(19,27,29,32)(20,25)(21,26)(22,24)(23,28,31,30);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,24)(23,31)(25,26)(27,32);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(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=gh,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 | #105901, #106007, #106009, … |
| maximal normal subgroup of | #105901, #106007, #106009, … |
| maximal quotient of | #160450, #160461, #160500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10564);