Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| 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 | |
437 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 437 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)(4,17)(5,7)(6,11)(8,15)(9,19)(10,20)(13,27)(14,25)(16,23)(18,22)(21,26)(24,28)(29,31)(30,32);(2,7)(4,18)(5,20)(8,23)(9,17)(10,11)(12,25)(13,28)(14,15)(19,21)(27,32)(29,30);(1,3)(2,8)(4,13)(5,14)(6,16)(7,12)(9,24)(10,23)(11,25)(15,20)(17,31)(18,28)(19,30)(21,29)(22,27)(26,32);(1,4,6,19)(2,9,11,17)(3,13,16,30)(5,21,20,18)(7,22,10,26)(8,24,25,31)(12,27,23,32)(14,29,15,28)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6632111, #6632112, #6632117, … |
| maximal normal subgroup of | #6632111, #6632112, #6632117, … |
| maximal quotient of | #332000, #332001, #332002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23426);