Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 663 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)(3,4)(5,6)(7,17)(8,22)(9,15)(10,14)(11,16)(12,19)(13,26)(18,27)(20,29)(21,28)(23,35)(24,31)(25,30)(32,34)(33,36);(5,7,11,21)(6,12,16,29)(8,23,24,18)(9,20,10,19)(13,27,30,35)(14,28,15,17)(22,32,31,36)(25,33,26,34);(2,4)(5,8,9,25,11,24,10,26)(6,13,14,31,16,30,15,22)(7,18,19,33,21,23,20,34)(12,27,17,32,29,35,28,36);(8,24)(13,30)(18,23)(22,31)(25,26)(27,35)(32,36)(33,34)Polycyclic presentation:
2^8|b^2=h,c^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #394601, #394634, #394635, … |
| maximal normal subgroup of | #394601, #394634, #394635, … |
| maximal quotient of | #238991, #238992, #238995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16520);