Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2668 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 441 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:
(1,2)(3,12)(4,8)(5,9)(6,11)(7,15)(10,20)(13,28)(14,22)(16,26)(17,27)(18,23)(19,25)(21,29)(24,31)(30,32);(1,3)(2,7)(4,17)(5,14)(6,16)(8,21)(9,12)(10,26)(11,22)(13,19)(15,20)(18,30)(23,27)(24,28)(25,32)(29,31);(1,4,6,19)(2,8,11,25)(3,13,16,17)(5,18,20,31)(7,21,22,32)(9,23,10,24)(12,27,26,28)(14,29,15,30);(2,9)(3,14)(5,20)(8,23)(10,11)(12,26)(13,29)(15,16)(17,30)(18,31)(24,25)(27,28)Polycyclic presentation:
2^8|c^2=h,e^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6792407, #6792408, #6792409, … |
| maximal normal subgroup of | #6792407, #6792408, #6792409, … |
| maximal quotient of | #331931, #331932, #331935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23424);