Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator, prime radical | C23 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,23,21,24,29,30,22,25,19,26,27,31,28,32);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,28,22,20,27,29)(18,23,24,30,25,26,31,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=fg,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,e^b=ef64 conjugacy classes — view the full character table.
| maximal subgroup of | #93002, #93006, #122383, … |
| maximal normal subgroup of | #93002, #93006, #122383, … |
| maximal quotient of | #124208, #124227, #124414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4482);