Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| 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 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
245 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 83 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)(5,6,10,15,9,13,11,26)(7,16,19,33,18,32,25,34)(8,20,23,28,22,36,24,14)(12,27,30,17,29,35,31,21);(5,7,8,21,11,19,24,27,9,18,22,17,10,25,23,35)(6,12,28,16,15,31,36,34,13,29,14,32,26,30,20,33);(1,2,3,4)(6,13)(12,29)(14,28)(15,26)(16,32)(20,36)(30,31)(33,34)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eh,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131862, #131863, #131870, … |
| maximal normal subgroup of | #131862, #131863, #131870, … |
| maximal quotient of | #140527, #140528, #140574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6202);