Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,12,10,13,8,16)(7,11,14,20,15,19,17,18);(7,14,15,17)(11,18,19,20);(2,4)(5,7)(6,11)(8,17)(9,14)(10,15)(12,20)(13,19)(16,18)Polycyclic presentation:
2^8|a^2=g,b^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^c=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11080, #357746, #358519, … |
| maximal normal subgroup of | C2×#11080, #357746, #358519, … |
| maximal quotient of | #115971, #115972, #115982, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4068);