Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| 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 | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#459 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
526 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)(6,9)(7,11)(13,14,17,20)(15,18,16,19);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,15,17,16)(14,18,20,19);(6,9)(10,12)(14,19)(15,16)(18,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #209696, #209702, #209706, … |
| maximal normal subgroup of | #209696, #209702, #209706, … |
| maximal quotient of | #218464, #218470, #218488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15139);