Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
264 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,15,13,14,11,16)(17,18)(19,25,27,22)(20,28)(21,24)(23,29)(26,30,32,31);(2,5)(3,8)(6,7)(10,14)(11,12)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(9,11,13,12)(10,15,14,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=e,c^2=fg,d^2=g,e^2=h,b^a=bd,c^a=cfg,d^a=df,c^b=ch,d^b=dg,e^b=efg46 conjugacy classes — view the full character table.
| maximal subgroup of | #98086, #98092, #98094, … |
| maximal normal subgroup of | #98086, #98092, #98094, … |
| maximal quotient of | #96293, #96759, #96928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 707);