Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2308 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | #3485 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
659 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 659 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,8)(5,17,18,9)(6,10)(12,19,26,21)(13,20)(15,28,29,22)(16,23)(24,27)(25,31)(30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,21)(9,20)(10,11)(15,25)(16,26)(17,24)(18,27)(19,23)(22,31)(28,30)(29,32);(1,4,6,16)(2,8,10,23)(3,12,14,26)(5,15,18,29)(7,19,11,21)(9,22,17,28)(13,25,27,32)(20,30,24,31);(1,5)(2,9)(3,13)(4,15)(6,18)(7,20)(8,22)(10,17)(11,24)(12,25)(14,27)(16,29)(19,30)(21,31)(23,28)(26,32)Polycyclic presentation:
2^8|a^2=g,c^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #135517, #139740, #139745, … |
| maximal normal subgroup of | #135517, #139740, #139745, … |
| maximal quotient of | #213721, #213723, #213725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14765);