Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,30,31,27);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,22)(23,31)(24,29)(25,32);(11,15)(13,16)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,4,5,3)(2,7,8,6)(18,22)(23,31)(24,29)(25,32)Polycyclic presentation:
2^8|a^2=eg,b^2=eh,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101548, #101549, #295324, … |
| maximal normal subgroup of | #101548, #101549, #295324, … |
| maximal quotient of | #314966, #314976, #314986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20674);