Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1437 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
452 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)(3,12,15,8)(4,16)(5,20)(6,11)(7,10)(9,18)(13,27,28,32)(14,23,21,24)(17,30)(19,25)(22,29,26,31);(1,3,6,14,7,15,5,21)(2,8,11,24,10,12,20,23)(4,13,19,29,17,28,18,31)(9,22,16,27,25,26,30,32);(1,4,5,18,7,17,6,19)(2,9,11,16,10,25,20,30)(3,13,21,31,15,28,14,29)(8,22,24,27,12,26,23,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,e^b=eh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10950, #354310, #355093, … |
| maximal normal subgroup of | C2×#10950, #354310, #355093, … |
| maximal quotient of | #110054, #110055, #110056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3938);