Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,8)(9,10,11,12)(13,14)(15,20)(16,19)(17,18);(1,2,3,4)(13,14,17,20,16,19,15,18);(6,8)(13,14,16,19)(15,20,17,18)Polycyclic presentation:
2^8|a^2=g,b^2=dh,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^c=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11124, #357793, #357795, … |
| maximal normal subgroup of | C2×#11124, #357793, #357795, … |
| maximal quotient of | #116984, #117002, #117010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4112);