Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
735 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 735 conjugacy classes of subgroups — too many to draw. Only its 247 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6,7,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,28)(22,30)(23,32)(25,31)(27,34)(29,33)(35,36)(37,39)(38,40);(1,2,3,4)(6,8)(9,11,13,24,15,25,14,23)(10,16,19,21,20,32,18,31)(12,22,27,37,29,38,28,36)(17,30,33,35,34,40,26,39);(9,12,13,27,15,29,14,28)(10,17,19,33,20,34,18,26)(11,22,24,37,25,38,23,36)(16,30,21,35,32,40,31,39);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,32,31)(17,26,34,33)(22,36,38,37)(30,35,40,39)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=fh,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124264, #124941, #140616, … |
| maximal normal subgroup of | #124264, #124941, #140616, … |
| maximal quotient of | #204265, #204282, #204299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14220);