Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #379 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | C22⋊#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,13,16,8)(4,18,20,26)(5,11)(6,12)(7,10)(9,19,25,21)(14,28)(15,24,17,23)(22,29)(27,31)(30,32);(1,3,6,17,7,16,5,15)(2,8,12,24,10,13,11,23)(4,14,19,29,20,30,21,31)(9,22,18,32,25,27,26,28);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,25)(11,26)(12,18)(13,27)(15,31)(16,30)(17,29)(23,28)(24,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,e^c=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10778, #354822, #355485, … |
| maximal normal subgroup of | C2×#10778, #354822, #355485, … |
| maximal quotient of | #105644, #105645, #105646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3766);