Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 32768 |
| 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 | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
522 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)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,3,6,2)(4,7,5,8)(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(2,3)(7,8)(10,15)(14,23)(16,20)(17,24);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^8|a^2=f,b^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 | #107489, #107493, #107726, … |
| maximal normal subgroup of | #107489, #107493, #107726, … |
| maximal quotient of | #162561, #162562, #162563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10489);