Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
597 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(1,2)(3,4)(5,6)(7,15,17,11)(8,19,20,12)(9,14)(10,13)(16,23)(18,28,31,24)(21,32,33,25)(22,26)(27,30)(29,34)(35,36);(2,4)(5,7,9,18,10,17,22,31)(6,11,14,24,13,15,26,28)(8,16,21,29,20,30,33,36)(12,23,25,34,19,27,32,35);(5,8)(6,12)(7,16)(9,21)(10,20)(11,23)(13,19)(14,25)(15,27)(17,30)(18,29)(22,33)(24,34)(26,32)(28,35)(31,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^b=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93480, #93481, #93486, … |
| maximal normal subgroup of | #93480, #93481, #93486, … |
| maximal quotient of | #105572, #105573, #105574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3752);