Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
307 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 135 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)(7,15)(8,12)(9,13)(10,14)(11,16)(17,20)(18,19)(21,22,25,29,26,30,24,28)(23,31,32,35,34,36,33,27);(2,4)(5,7,9,17,10,18,8,16)(6,11,12,19,14,20,13,15)(21,23)(22,27,30,35)(24,33)(25,32)(26,34)(28,36,29,31);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,31,35,36)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,d^a=dg,f^a=fg,c^b=ch76 conjugacy classes — view the full character table.
| maximal subgroup of | #105777, #107258, #107444, … |
| maximal normal subgroup of | #105777, #107258, #107444, … |
| maximal quotient of | #94886, #94900, #94907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 882);