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) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
339 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,24)(17,20)(18,23)(19,25)(22,27)(26,28);(6,10)(7,8)(11,12)(13,15)(14,18,20,26)(16,23)(17,22)(19,21,24,27)(25,28);(5,7,9,8)(6,11,12,10)(15,22)(18,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=g,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 | #107301, #107302, #107303, … |
| maximal normal subgroup of | #107301, #107302, #107303, … |
| maximal quotient of | #95198, #95202, #95220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1005);