Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| 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 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
509 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)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(1,4,6,5)(2,7,3,8);(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=g,e^2=h,f^2=gh,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #106057, #107294, #107606, … |
| maximal normal subgroup of | #106057, #107294, #107606, … |
| maximal quotient of | #160533, #160541, #160552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10725);