Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(10,14)(11,15)(12,18)(16,20)(17,24)(19,23)(21,22);(9,10,15,22,12,17,19,20,13,16,23,24,18,21,11,14);(1,2,3,4)(10,16)(14,20)(17,21)(22,24)Polycyclic presentation:
2^8|a^2=f,b^2=dg,c^2=e,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #136334, #136335, #136342, … |
| maximal normal subgroup of | #136334, #136335, #136342, … |
| maximal quotient of | #135727, #135750, #135763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5896);