Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | Aut(G) = AGL(1,ℤ/8)×#442 (order 2048) · Inn(G) = D4×D8 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 105 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:
(2,8)(3,13)(4,17)(5,6)(9,10)(11,27)(12,26)(14,30)(15,31)(16,22)(18,23)(20,21)(24,25)(28,29);(1,2)(3,14)(4,18)(5,11)(6,23)(7,12)(8,17)(9,16)(10,15)(13,28)(19,29)(20,26)(21,27)(22,25)(24,32)(30,31);(1,3)(2,9)(4,15)(5,22)(6,13)(7,16)(8,24)(10,27)(11,28)(12,14)(17,19)(18,30)(20,32)(21,31)(23,25)(26,29);(1,3)(2,10)(4,19)(5,13)(6,22)(7,16)(8,25)(9,27)(11,29)(12,30)(14,18)(15,17)(20,31)(21,32)(23,24)(26,28)Polycyclic presentation:
2^8|e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gh,c^b=ch,d^b=de,e^b=eg,g^b=gh,f^c=fh,e^d=eg,g^d=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #492528, #492529, #492530, … |
| maximal normal subgroup of | #492528, #492529, #492530, … |
| maximal quotient of | #491731, #491732, #491737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26844);