Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C2×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10)(11,19)(12,18)(13,16)(14,17)(15,20)(21,24)(22,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,23)(18,24);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,19,23,24);(10,17)(15,23)(16,18)(19,24)Polycyclic presentation:
2^8|a^2=f,c^2=g,e^2=g,f^2=h,b^a=be,c^a=cg,d^a=dgh,e^a=eg,c^b=ch,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #107376, #107378, #107864, … |
| maximal normal subgroup of | #107376, #107378, #107864, … |
| maximal quotient of | #162550, #162551, #162552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10490);