Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C22⋊C4 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C2 | C23×C22⋊C4 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2406 conjugacy classes of subgroups; only the 989 normal subgroups are shown.
This group has 2406 conjugacy classes of subgroups — too many to draw. Only its 989 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 — 5 generators on 32 points:
(1,2,6,12,7,11,5,10)(3,13,15,26,16,8,20,22)(4,9,19,25,17,23,18,24)(14,27,29,32,28,21,30,31);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,22)(11,13)(12,26)(17,28)(18,29)(19,30)(23,27)(24,31)(25,32);(1,4)(2,9)(3,14)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(13,27)(15,29)(16,28)(20,30)(22,31)(26,32);(4,17)(9,23)(14,28)(18,19)(21,27)(24,25)(29,30)(31,32);(1,5,7,6)(2,10,11,12)(3,15,16,20)(4,18,17,19)(8,22,13,26)(9,24,23,25)(14,29,28,30)(21,31,27,32)Polycyclic presentation:
2^8|a^2=g,e^2=h,g^2=h,b^a=bf,f^a=fh,g^b=gh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #60076⋊C3 |
| maximal subgroup of | #146589, #355926, #355934, … |
| maximal normal subgroup of | #146589, #355926, #355934, … |
| maximal quotient of | #6342543, #6342544, #6342545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53064);