Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(5,6,11,16)(7,17,20,31)(8,21,24,13)(9,15,10,14)(12,25,28,19)(18,29,34,35)(22,27,33,36)(23,30,26,32);(1,2)(3,4)(5,7,11,20)(6,12,16,28)(8,22,24,33)(9,25,10,19)(13,29,21,35)(14,31,15,17)(18,26,34,23)(27,32,36,30);(2,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,27)(14,17)(15,31)(16,28)(21,36)(22,23)(24,34)(26,33)(29,30)(32,35);(5,8,10,26,11,24,9,23)(6,13,15,32,16,21,14,30)(7,18,25,33,20,34,19,22)(12,27,31,35,28,36,17,29)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,e^b=eh,f^c=fh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151261, #151263, #151429, … |
| maximal normal subgroup of | #151261, #151263, #151429, … |
| maximal quotient of | #339712, #339714, #339728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23858);