Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
240 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,11,18,14,23,10,22,13,19,9,16,12,7,20,17)(8,24,27,15,31,33,26,21,29,36,25,35,28,32,30,34);(1,2,3,4)(5,7,13,23)(6,14,19,12)(8,21,29,34)(9,18,11,16)(10,22,20,17)(15,28,35,31)(24,30,36,26)(25,32,27,33);(5,8,13,29)(6,15,19,35)(7,21,23,34)(9,26,11,30)(10,25,20,27)(12,31,14,28)(16,36,18,24)(17,33,22,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=h,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=egh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13018, #494517, #494519, … |
| maximal normal subgroup of | C2×#13018, #494517, #494519, … |
| maximal quotient of | #136690, #136724, #136899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6006);