Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2366 |
| prime radical | #3147 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,14,18,27)(4,19)(5,11)(6,12)(7,24)(8,13)(9,17,28,23)(10,20)(15,25,16,29)(21,30)(22,32,31,26);(1,3,6,23,8,18,7,17)(2,9,12,27,13,28,24,14)(4,15,5,22,21,16,20,31)(10,25,19,32,11,29,30,26);(1,4,7,20,8,21,6,5)(2,10,12,19,13,11,24,30)(3,15,17,31,18,16,23,22)(9,25,27,32,28,29,14,26);(1,5,6,21,8,20,7,4)(2,11,12,30,13,10,24,19)(3,16,17,22,18,15,23,31)(9,26,14,29,28,32,27,25)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136490, #136492, #136509, … |
| maximal normal subgroup of | #136490, #136492, #136509, … |
| maximal quotient of | #231466, #231467, #231468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16184);