Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#537 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
869 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 869 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,15,11,14,9,13)(7,8,12,17,16,18,20,19);(5,7,10,20,11,16,9,12)(6,8,13,17,14,18,15,19);(2,4)(5,8)(6,12)(7,13)(9,19)(10,17)(11,18)(14,20)(15,16);(5,9,11,10)(6,13,14,15)(7,12,16,20)(8,17,18,19)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #364633, #364634, #365074, … |
| maximal normal subgroup of | #364633, #364634, #365074, … |
| maximal quotient of | #195929, #195930, #195931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13980);