Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,18)(17,20);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(14,18)(15,16)(19,20);(1,3,4,2)(13,15,17,16)(14,19,20,18)Polycyclic presentation:
2^8|a^2=g,c^2=e,d^2=gh,f^2=h,b^a=be,c^a=cf,f^a=fh,d^c=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100224, #100232, #208866, … |
| maximal normal subgroup of | #100224, #100232, #208866, … |
| maximal quotient of | #212019, #212031, #212061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14567);