Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10,13,16,14,17,12,21)(11,18,19,23,20,15,22,24);(2,6)(3,4)(7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24);(2,7)(6,8);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)Polycyclic presentation:
2^8|a^2=f,d^2=g,e^2=h,f^2=gh,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #106204, #107389, #107478, … |
| maximal normal subgroup of | #106204, #107389, #107478, … |
| maximal quotient of | #160571, #160595, #160611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10732);