Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#454 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 167 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 — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16)(7,17,20,33)(8,13,24,32)(9,15,10,14)(12,25,28,19)(18,35,22,29)(21,30,36,27)(23,34,26,31);(1,3,4,2)(5,7)(6,12)(8,21)(9,25)(10,19)(11,20)(13,29)(14,33)(15,17)(16,28)(18,23)(22,26)(24,36)(27,31)(30,34)(32,35);(5,7)(6,12)(8,22)(9,19)(10,25)(11,20)(13,30)(14,17)(15,33)(16,28)(18,24)(21,26)(23,36)(27,32)(29,34)(31,35);(5,8)(6,13)(7,18)(9,26)(10,23)(11,24)(12,27)(14,34)(15,31)(16,32)(17,35)(19,36)(20,22)(21,25)(28,30)(29,33)Polycyclic presentation:
2^8|a^2=g,b^2=gh,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,d^c=dh,f^c=fh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #152569, C2×#22895, #6381471, … |
| maximal normal subgroup of | #152569, C2×#22895, #6381471, … |
| maximal quotient of | #226939, #226947, #226955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15883);