Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1436 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,7,12,6,10)(3,13,15,8,17,23,16,24)(4,18,20,25,21,26,19,9)(14,28,30,27,31,22,29,32);(1,3,6,16,7,17,5,15)(2,8,11,23,12,24,10,13)(4,14,19,29,21,31,20,30)(9,22,18,32,25,28,26,27);(1,4,5,20,7,21,6,19)(2,9,11,18,12,25,10,26)(3,14,15,30,17,31,16,29)(8,22,23,32,24,28,13,27)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,e^b=eh,d^c=dh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10906, #354322, #355278, … |
| maximal normal subgroup of | C2×#10906, #354322, #355278, … |
| maximal quotient of | #109536, #109537, #109540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3894);