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 | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,24)(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);(2,3)(7,8)(10,16)(15,23)(17,18)(19,24);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,17,16,18)(11,20,21,22)(15,19,23,24)Polycyclic presentation:
2^8|a^2=f,d^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 | #107490, #107491, #107921, … |
| maximal normal subgroup of | #107490, #107491, #107921, … |
| maximal quotient of | #162566, #162567, #162568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10491);