Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | Aut(G) = C2×#6346637 (order 1024) · Inn(G) = #3109 · Out(G) = C23 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #2190 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
263 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 93 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)(3,8)(4,14)(5,18)(6,22)(7,11)(9,17)(10,19)(12,29)(13,28)(15,27)(16,31)(20,23)(21,26)(24,30)(25,32);(1,3,7,6)(2,8,11,22)(4,12,16,15)(5,19,21,13)(9,23,24,25)(10,26,28,18)(14,27,31,29)(17,32,30,20);(1,4)(2,9)(3,12)(5,20)(6,15)(7,16)(8,23)(10,27)(11,24)(13,30)(14,18)(17,19)(21,32)(22,25)(26,31)(28,29);(4,15,16,12)(5,21)(9,23,24,25)(10,28)(13,19)(14,27,31,29)(17,20,30,32)(18,26)Polycyclic presentation:
2^8|b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #485583, #485584, #485588, … |
| maximal normal subgroup of | #485583, #485584, #485588, … |
| maximal quotient of | #494053, #494997, #494998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26678);