Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #396 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,6,11,8,10,13,9,19)(7,14,15,12,17,18,16,20);(5,7,11,15,10,17,9,16)(6,12,8,18,13,20,19,14);(5,8,9,13,10,19,11,6)(7,12,15,14,17,20,16,18)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #11312⋊C3 |
| maximal subgroup of | C2×#11312, #358327, C4×#2069, … |
| maximal normal subgroup of | C2×#11312, #358327, C4×#2069, … |
| maximal quotient of | #121067, #121100, #121101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4300);