Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #394 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,11,16,23)(8,12,19,26)(9,14)(10,13)(15,22)(17,24,30,35)(18,25)(20,27,32,33)(21,28)(29,34)(31,36);(5,7)(6,11,13,23)(8,18,19,31)(9,17)(10,16)(12,25)(14,24,28,35)(15,20,29,32)(21,30)(22,33)(26,36)(27,34);(5,8)(6,12,13,26)(7,15,16,29)(9,20)(10,19)(11,22)(14,27,28,33)(17,18,30,31)(21,32)(23,34)(24,25)(35,36)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,g^b=gh,e^c=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93942, #136191, #136192, … |
| maximal normal subgroup of | #93942, #136191, #136192, … |
| maximal quotient of | #120617, #120619, #120620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4273);