Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
283 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20)(21,22)(23,30)(24,28)(25,34)(26,29)(27,32)(31,36)(33,35);(5,7)(6,10,12,18)(8,15)(9,14)(11,13,16,19)(17,20)(21,23,25,33,26,32,24,31)(22,27,28,35,29,30,34,36);(7,14)(10,18)(13,19)(15,20)(21,24,26,25)(22,28,29,34)(23,31,32,33)(27,35,30,36)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=g,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #107191, #107192, #107218, … |
| maximal normal subgroup of | #107191, #107192, #107218, … |
| maximal quotient of | #95285, #95287, #95290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1020);