Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
661 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 661 conjugacy classes of subgroups — too many to draw. Only its 263 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 — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,9,24)(7,16,20,26,18,30,19,12)(8,13,23,28,21,27,22,34)(17,29,32,35,33,36,31,25);(5,7,9,19,11,18,10,20)(6,12,14,16,15,26,24,30)(8,17,22,31,21,33,23,32)(13,25,28,29,27,35,34,36);(2,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,21)(12,25)(14,28)(15,27)(16,29)(18,33)(19,31)(20,32)(24,34)(26,35)(30,36);(7,18)(8,21)(12,26)(13,27)(16,30)(19,20)(22,23)(28,34)Polycyclic presentation:
2^8|a^2=g,b^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #135328, #135329, #135348, … |
| maximal normal subgroup of | #135328, #135329, #135348, … |
| maximal quotient of | #188712, #188719, #188726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13657);