Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 474 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) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22)(23,31)(24,27)(25,29)(26,35)(28,33)(30,34)(32,36);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,25,24)(22,27,29,31)(26,32,34,33)(28,30,36,35);(2,4)(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,24,25,23)(22,28,29,36)(26,32,34,33)(27,30,31,35);(8,16)(11,18)(14,20)(17,19)(23,24)(27,31)(28,36)(32,33)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #249981, #249985, #249998, … |
| maximal normal subgroup of | #249981, #249985, #249998, … |
| maximal quotient of | #327878, #327883, #327886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22637);