Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #399 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,10,14,11,15,9,17)(7,16,19,20,13,8,18,12);(1,2)(3,4)(5,7,9,18,11,13,10,19)(6,12,14,16,15,20,17,8);(2,4)(5,8,9,20,11,16,10,12)(6,13,14,18,15,7,17,19)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=e,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #11338⋊C3 |
| maximal subgroup of | #93953, #136138, #136268, … |
| maximal normal subgroup of | #93953, #136138, #136268, … |
| maximal quotient of | #121328, #121344, #121345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4326);