Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #340 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| prime radical | C23 | #103 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,21,24,22,26,31,25)(19,20,27,29,23,30,28,32);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,20,25,21,28,29,26,22,23,30,24,31,27,32,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,d^2=h,e^2=f,f^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,c^b=ch,d^b=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7223, #127272, #127273, … |
| maximal normal subgroup of | C2×#7223, #127272, #127273, … |
| maximal quotient of | #93310, #93349, #93364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 211);