Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C22⋊C4 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C22×C22⋊C8 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 261 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:
(1,2,4,9,5,10,15,22,6,7,14,18,16,19,26,29)(3,8,12,20,13,21,25,30,11,17,23,27,24,28,31,32);(2,7)(4,14)(8,17)(10,19)(12,23)(15,26)(21,28)(25,31);(1,3)(2,8)(4,12)(5,13)(6,11)(7,17)(9,20)(10,21)(14,23)(15,25)(16,24)(18,27)(19,28)(22,30)(26,31)(29,32);(3,11)(8,17)(12,23)(13,24)(20,27)(21,28)(25,31)(30,32)Polycyclic presentation:
2^8|a^2=f,f^2=g,g^2=h,b^a=be,e^a=eh,f^b=fh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #33337⋊C3 |
| maximal subgroup of | #137702, #137703, #137748, … |
| maximal normal subgroup of | #137702, #137703, #137748, … |
| maximal quotient of | #480008, #480044, #480061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26325);