Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
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 | |
234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 234 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)(10,14,15,16)(17,18,21,25,22,26,20,24)(19,27,30,23,28,32,29,31);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,30)(21,29)(22,28)(24,32)(25,27)(26,31);(9,11,13,12)(10,14,15,16)(19,28)(23,31)(27,32)(29,30)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,d^2=g,e^2=h,b^a=bd,d^a=df,c^b=ch,d^b=dg,e^b=efg58 conjugacy classes — view the full character table.
| maximal subgroup of | #97806, #97807, #97808, … |
| maximal normal subgroup of | #97806, #97807, #97808, … |
| maximal quotient of | #96271, #96744, #96745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 659);