Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C22×#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
998 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 998 conjugacy classes of subgroups — too many to draw. Only its 557 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)(3,11,13,7)(4,8)(5,10)(6,9)(12,23,25,18)(14,24,27,19)(15,20)(16,21)(17,22)(26,31,32,29)(28,30);(1,3,5,14,6,13,17,27)(2,7,10,19,9,11,22,24)(4,12,16,26,15,25,28,32)(8,18,21,29,20,23,30,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,18)(9,20)(10,21)(11,23)(13,25)(14,26)(17,28)(19,29)(22,30)(24,31)(27,32);(4,15)(8,20)(12,25)(16,28)(18,23)(21,30)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=f,b^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 | #20360⋊C3 |
| maximal subgroup of | #355478, #355485, #355490, … |
| maximal normal subgroup of | #355478, #355485, #355490, … |
| maximal quotient of | #181052, #181056, #181060, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13348);