Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7426 |
| automorphism | Aut(G) = #72231851 (order 1024) · Inn(G) = #1351 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1351 |
| commutator | C2×C22⋊C4 | C2×C4 |
| Frattini, cosocle | #381 | C22 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,9,6,10,19,26)(3,11,13,27,14,28,30,32)(4,15,17,31,18,8,23,24)(7,20,21,12,22,29,25,16);(1,3)(2,7)(4,16,18,12)(5,13)(6,14)(8,11,15,28)(9,25)(10,22)(17,29,23,20)(19,30)(21,26)(24,27,31,32)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,c^b=cf,d^b=defg,f^b=fh,g^b=gh,d^c=dg,e^c=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7427, #150261, #150395, … |
| maximal normal subgroup of | C2×#7427, #150261, #150395, … |
| maximal quotient of | #93239, #93240, #93943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 415);