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) = C2×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 | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
250 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 65 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)(3,7,8,5)(4,6)(9,10,12,14,13,16,11,15)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3,4,8)(2,5,6,7)(10,14)(11,12)(15,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(9,11,13,12)(10,14,16,15)(18,23)(22,30)(24,31)(25,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=g,e^2=h,b^a=bd,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=efg46 conjugacy classes — view the full character table.
| maximal subgroup of | #98057, #98061, #98064, … |
| maximal normal subgroup of | #98057, #98061, #98064, … |
| maximal quotient of | #96278, #96749, #96932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 693);