Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#120 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#385 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 237 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:
(1,2,5,10,6,9,16,22)(3,7,13,19,12,18,26,30)(4,8,15,21,14,20,27,31)(11,23,25,32,24,17,28,29);(1,3,5,13,6,12,16,26)(2,7,10,19,9,18,22,30)(4,11,15,25,14,24,27,28)(8,17,21,29,20,23,31,32);(1,4)(2,8)(3,11)(5,15)(6,14)(7,17)(9,20)(10,21)(12,24)(13,25)(16,27)(18,23)(19,29)(22,31)(26,28)(30,32);(2,9)(7,18)(8,20)(10,22)(17,23)(19,30)(21,31)(29,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh,f^b=fh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #148247, #148248, #148250, … |
| maximal normal subgroup of | #148247, #148248, #148250, … |
| maximal quotient of | #192028, #192029, #192032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13880);