Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | Aut(G) = #72338915 (order 4096) · Inn(G) = #2668 · Out(G) = C22×D4 |
| 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 | |
451 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 451 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)(3,13)(4,10)(5,8)(6,11)(7,12)(9,14)(15,29)(16,21)(17,26)(18,22)(19,25)(20,24)(23,30)(27,31)(28,32);(2,8)(3,14)(4,17)(5,6)(10,25)(11,12)(13,21)(16,20)(18,19)(23,29)(27,32)(28,30);(1,3)(2,9)(4,15)(5,14)(6,20)(7,16)(8,21)(10,23)(11,13)(12,24)(17,30)(18,28)(19,31)(22,29)(25,32)(26,27);(1,4)(2,10)(3,15)(5,18)(6,17)(7,19)(8,22)(9,23)(11,26)(12,25)(13,27)(14,28)(16,31)(20,30)(21,29)(24,32)Polycyclic presentation:
2^8|e^2=h,f^2=h,g^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,g^b=gh,f^c=fh,g^c=gh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151318, #151320, #151322, … |
| maximal normal subgroup of | #151318, #151320, #151322, … |
| maximal quotient of | #331927, #331928, #331929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23429);