Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
193 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,22,25,29)(23,26,31,36)(24,27,33,28)(30,34,32,35);(1,2)(3,4)(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,23,25,31)(22,26,29,36)(24,32,33,30)(27,35,28,34);(1,3,4,2)(6,12)(7,15)(8,9)(13,14)(16,17)(18,19)(22,27)(24,33)(26,34)(28,29)(30,32)(35,36)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=d,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94681, C2×#10428, #238560, … |
| maximal normal subgroup of | #94681, C2×#10428, #238560, … |
| maximal quotient of | #104840, #104844, #104845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3416);