Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| 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 | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 75 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)(5,6,9,15,23,31,24,33,11,13,8,14,10,16,26,35)(7,17,20,30,27,34,22,12,19,29,18,36,21,32,25,28);(1,2,3,4)(5,7)(6,12)(8,22)(9,25)(10,27)(11,19)(13,28)(14,32)(15,34)(16,36)(17,33)(18,24)(20,26)(21,23)(29,35)(30,31);(6,13)(8,9)(12,28)(16,31)(18,20)(22,25)(24,26)(30,36)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #131323, #131324, #131328, … |
| maximal normal subgroup of | #131323, #131324, #131328, … |
| maximal quotient of | #136610, #136638, #136884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5988);