Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3,6,4,5,7,8)(9,10,12,15,13,16,11,14)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(2,5)(3,7)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(9,11,13,12)(10,14,16,15)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=fgh,d^2=g,e^2=h,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg52 conjugacy classes — view the full character table.
| maximal subgroup of | #94141, #94388, #96982, … |
| maximal normal subgroup of | #94141, #94388, #96982, … |
| maximal quotient of | #96237, #96728, #96923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 593);